Basic Properties of the Polygonal Sequence

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Department of Physics and Engineering Physics, Obafemi Awolowo University, 220005 Ile-Ife, Nigeria

2010 Mathematics Subject Classiﬁcation: Primary 11B99; Secondary 11H99. Keywords: polygonal number, triangular number, summation identity, generating function, partial sum, recurrence relation.

Abstract We study various properties of the polygonal numbers; such as their recurrence relations, fundamental identities, weighted binomial and ordinary sums and the partial sums and generating functions of their powers. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers as is the case in most literature.

1 Introduction

For a ﬁxed integer r ≥ 2, the polygonal sequence (Pn,r) is deﬁned through the recurrence relation, Pn,r = 2Pn−1,r − Pn−2,r + r − 2 (n ≥ 2) ,

with P0,r = 0,P1,r = 1. Thus, the polygonal sequence is a second order, linear non-homogeneous sequence with constant coefficients. In § 1.2 we shall see that the polygonal numbers can also be defined by a third order homogeneous recurrence relation. The fixed number r is called the order or rank of the polygonal sequence. The nth polygonal number of order r, Pn,r, derives its name from representing a polygon by dots in a plane: r is the number of sides and n is the number of dots on each side. Thus, for a triangular number, r = 3, for a square number, r = 4 (four sides), and so on. The book by Deza and Deza [3] is an excellent source book on polygonal numbers and figurate numbers in general. Cook and Bacon [2] derived some summation formulas involving the polygonal numbers. Hoggatt and Bicknell [7] studied the triangular numbers. The ar- ticle by Garge and Shirali [5] showcases some interesting basic properties of the triangular numbers. We refer the reader to Dickson [4, Chapter I] for an historical perspective on polygonal numbers.

In the sequel, the equations look nicer and simpler if we characterize polygonal numbers by a quantity γ(r) deﬁned by γ(r) ≡ r − 2. For brevity we will always write γ when we mean γ(r). The recurrence relation for polygonal numbers is then,

Pn,r = 2Pn−1,r − Pn−2,r + γ (n ≥ 2) , (1.1)

with P0,r = 0,P1,r = 1.

We have γ = 0 for natural numbers (Pn,2 = n), γ = 1 for triangular numbers, γ = 2 for square numbers and so on. The diﬀerence equation (1.1) is readily solved, yielding an exact formula for the nth r−gonal number, namely, γ (γ − 2) P = n2 − n . (1.2) n,r 2 2

The nth triangular number, Pn,3, will, henceforth, be denoted by Tn:

n(n + 1) T = P = . n n,3 2 By re-arranging the rhs of (1.2), we see that every polygonal number can be expressed in terms of a triangular number: Pn,r = n + γ Tn−1 . (1.3)

1.1 Extension to negative subscripts As is the case with other integer sequences, it is possible to continue the polygonal sequence to include negative indices. Write the recurrence relation as

Pn,r = 2Pn+1,r − Pn+2,r + γ

and replace n with −n, obtaining

P−n,r = 2P−(n−1),r − P−(n−2),r + γ ,

for n = 1, 2,...

This gives P−1,r = γ − 1, P−2,r = 3γ − 2, P−3,r = 6γ − 3, and so on. The solution of the diﬀerence equation

Pm,r = 2Pm+1,r − Pm+2,r + γ ,

with initial conditions P−1,r = γ − 1, P−2,r = 3γ − 2, is

γ (γ − 2) P = m2 − m . m,r 2 2 Writing −n for m we have γ (γ − 2) P = n2 + n . (1.4) −n,r 2 2

Thus, the sequence (Pn,r) is deﬁned for all integers; with the exact values given by (1.2) and (1.4).

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Note, in particular, that, n2 −n (n − 1)n T = P = + = = T . (1.5) −n −n,3 2 2 2 n−1 Addition of (1.2) and (1.4) gives

2 P−n,r = γn − Pn,r , (1.6) while their diﬀerence gives P−n,r = (γ − 2)n + Pn,r . (1.7) Writing −n for n in (1.3) and making use of (1.5), we also have

P−n,r = −n + γ Tn . (1.8)

1.2 A third order linear recurrence relation Write the polygonal recurrence relation (1.1) in two ways,

Pj,r − 2Pj−1,r = γ − Pj−2,r , (1.9)

2Pj,r − Pj−1,r = Pj+1,r − γ , (1.10) and add to obtain 3Pj,r − 3Pj−1,r = Pj+1,r − Pj−2,r , which can be written Pj,r = 3Pj−1,r − 3Pj−2,r + Pj−3,r . (1.11) Relation (1.11) is a special case of a more general recurrence relation, given in Corollary 5. Note that (1.11) with the seeds P0,r = 0, P1,r = 1 and P2,r = γ + 2 makes (Pn,r) a third order recurrence sequence.

1.3 Sum of the ﬁrst k polygonal numbers Theorem 1. If k is an integer, then,

k X γ P = (k − 1) + 1 T , (1.12) j,r 3 k j=0

γ γ−2 k  2 Tk − 4 k, if k is even ; X j  (−1) Pj,r = (1.13) γ γ−2 j=0  − 2 Tk + 4 (k + 1), if k is odd . Proof. Using (1.2), we have

k k k X γ X γ − 2 X P = j2 − j , j,r 2 2 j=0 j=0 j=0

k k k X γ X γ − 2 X (−1)jP = (−1)jj2 − (−1)jj , j,r 2 2 j=0 j=0 j=0

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from which identities (1.12) and (1.13) follow upon using the following well-known results:

k k X X k(k + 1)(2k + 1) (2k + 1) j = T , j2 = = T , k 6 3 k j=0 j=0

  k k/2, if k is even, k T , if k is even, X  X  k (−1)jj = , (−1)jj2 = j=0  −(k + 1)/2, if k is odd, j=0  −Tk, if k is odd.

We remark that, since,

k k X 1 − e(k+1)x X 1 + (−1)ke(k+1)x exj = , (−1)jexj = , 1 − ex 1 + ex j=0 j=0 the sums of powers of consecutive integers are readily found from

k i (k+1)x k i k (k+1)x X i d 1 − e X j i d 1 + (−1) e j = , (−1) j = . (1.14) dxi 1 − ex dxi 1 + ex j=0 x=0 j=0 x=0

1.4 Some fundamental identities What Garge and Shirali [5] noted about triangular numbers is true about polygonal numbers in general: it is generally relatively easy to prove relations and identities involving polygonal numbers, often by simple algebra; the real challenge is in discovering them. By using the exact formula (1.2), it is straightforward to verify the following identities involving the polygonal numbers:

Pjm,r = Pj,rPm,r − γ(γ − 2)Tm−1Tj−1 , (1.15)

Pm+n,r − Pm−n,r = n(Pm+1,r − Pm−1,r) , (1.16)

2 Pm+n,r + Pm−n,r = 2Pm,r + γn , (1.17)

Pm+n,rPm−n,r = (Pm,r − Pn,r)(Pm,r − P−n,r) , (1.18)

Pn+m,r = (m + 1)Pn,r − mPn−1,r + γTm , (1.19)

2 Pn+jm,r = Pn,r + jPm,r + γm Tj−1 + γmnj , (1.20)

2 Pn−jm,r = Pn,r − jPm,r + γm Tj − γmnj . (1.21) In the particular case of triangular numbers (γ = 1), we have

Tjm = TjTm + Tm−1Tj−1 , (1.22)

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Tm+n − Tm−n = n(Tm+1 − Tm−1) , (1.23)

2 Tm+n + Tm−n = 2Tm + n , (1.24)

Tn+mTn−m = (Tn − Tm)(Tn − Tm−1) , (1.25)

Tn+m = (m + 1)Tn − mTn−1 + Tm , (1.26)

2 Tmj+n = Tn + jTm + m Tj−1 + mnj , (1.27)

2 Tn−mj = Tn − jTm + m Tj − mnj . (1.28) Note that the well known identity:

Tm+n = Tn + Tm + mn , (1.29)

is a special case of (1.27).

1.5 Basic summation tools In the sequel, we will often make use of the index shift identity,

k k a−1 k+a X j −a X j −a X j −a X j fj+ax = x fjx − x fjx + x fjx , (1.30) j=0 j=0 j=0 j=k+1

where (fi) is any real sequence and a and k are any integers. Note that if a < 1, then, a−1 −1 X j X j fjx ≡ − fjx , (1.31) j=0 j=a and k+a k X j X j fjx ≡ − fjx . (1.32) j=k+1 j=k+a+1 We also require the following telescoping summation identities:

k X (fj+1 − fj) = fk+1 − f0 , (1.33) j=0

k X j k (−1) (fj+1 + fj) = (−1) fk+1 + f0. (1.34) j=0 As an immediate application of (1.33), we can extract a summation identity from the recurrence relation (1.10).

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Theorem 2. If k is an integer, then,

k X j k+1 2 Pj,r = 2 (Pk−1,r + γ) + 2(1 − 2γ) . j=0 Proof. Write identity (1.10) as

Pj,r − γ = 2Pj−1,r − Pj−2,r and multiply through by 2j−1 to obtain

j−1 j−1 j j−1 2 Pj,r − γ2 = 2 Pj−1,r − 2 Pj−2,r . Thus, k k k X j−1 X j−1 X j j−1 2 Pj,r − γ 2 = (2 Pj−1,r − 2 Pj−2,r) . (1.35) j=0 j=0 j=0 j−1 To evaluate the sum on the right hand side, identify fj = 2 Pj−2,r and use identity (1.33) to obtain k X 1 (2jP − 2j−1P ) = 2kP − (3γ − 2) . j−1,r j−2,r k−1,r 2 j=0 Identity (1.35) now reads

k k X X 1 2j−1P − γ 2j−1 = 2kP − (3γ − 2) j,r k−1,r 2 j=0 j=0 or k k X j X j k+1 2 Pj,r − γ 2 = 2 Pk−1,r − 3γ + 2 , j=0 j=0 Pk j k+1 from which the result follows when we insert the geometric sum j=0 2 = 2 − 1.

2 A master identity

Theorem 3. Let a, b, c, d and e be integers. Then,

(Ta−cTc−bTb−a + Tb−cTc−aTa−b)Pd+e,r

= (Tb−cTc−aTe−b − Tb−cTc−bTe−a + Tc−bTb−aTe−c)Pd+a,r

+ (Ta−cTc−bTe−a − Ta−cTc−aTe−b + Tc−aTa−bTe−c)Pd+b,r

+ (Tb−cTa−bTe−a − Ta−bTb−aTe−c + Ta−cTb−aTe−b)Pd+c,r . Proof. We seek to express a polygonal number as a linear combination of three triangular numbers. Let Pd+e,r = f1Te−a + f2Te−b + f3Te−c , (2.1)

where a, b, c, d and e are arbitrary integers and the coeﬃcients f1, f2 and f3 are to be determined. Setting e = a, e = b and e = c, in turn, we obtain three simultaneous equations:

Pd+a,r = f2Ta−b + f3Ta−c ,Pd+b,r = f1Tb−a + f3Tb−c ,Pd+c,r = f1Tc−a + f2Tc−b .

The identity of Theorem 3 is established by solving these equations for f1, f2 and f3 and substituting the solutions into identity (2.1).

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Note that the identity of Theorem 3 can also be written

(a − b)(a − c)(b − c)Pd+e,r = (b − c)(e − c)(e − b)Pd+a,r

− (a − c)(e − a)(e − c)Pd+b,r (2.2)

+ (a − b)(e − a)(e − b)Pd+c,r . Theorem 3 yields a myriad of recurrence relations, of which we can mention a couple. Corollary 4. Let m and j be integers. Then

Pj+m,r = Tm−1Pj+2,r + (Tm − 3Tm−1)Pj+1,r + Tm−2Pj,r .

Proof. Set c = 0, b = 1, a = 2, d = j, e = m in Theorem 3. In particular, we have the addition formula for the triangular numbers:

Tj+m = Tm−1Tj+2 + (Tm − 3Tm−1)Tj+1 + Tm−2Tj . (2.3)

Corollary 5. Let m and j be integers. Then,

TmPj+1,r = Tm+1Pj,r − Tm+1Pj−m,r + TmPj−m−1,r .

Proof. Set a = 0, e = 1, b = −m, c = −m − 1 and d = j in (2.2). By writing −m for m, the above identity can also be written

Tm−1Pj+1,r = Tm−2Pj,r − Tm−2Pj+m,r + Tm−1Pj+m−1 . (2.4)

In particular, we have

TmTj+1 = Tm+1Tj − Tm+1Tj−m + TmTj−m−1 . (2.5)

Corollary 6. Let j, a and e be integers. Then,

(j + a)(a − c)(j + c)Pj+e,r = (j + c)(e − c)(j + e)Pj+a,r

− (j + a)(e − a)(j + e)Pj+c,r . Proof. Set d = −b followed by b = −j in identity 2.2. Setting n = 0 in Corollary 6 gives

(cTa − aTc)Pe,r = (cTe − eTc)Pa,r + (eTa − aTe)Pc,r . (2.6)

3 Linear properties: partial sums and generating functions

In this section, we will derive expressions for the partial sum of the polygonal numbers, Pk j P∞ j namely, j=0 Pj,rx and hence obtain the generating function, j=0 Pj,rx . Two equivalent expressions will be derived, one based on identity (1.10), the non homogeneous second order recurrence relation and the other based on identity (1.11), the homogeneous third order recurrence relation for polygonal numbers. We will also derive an expression for the partial Pk j sum of the polygonal numbers with indices in arithmetic progression, j=0 Pmj+n,rx .

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3.1 Partial sum of polygonal numbers Theorem 7. If k is an integer, then,

k X x γx P xj = xk(xP − P ) + 1 − γ + (1 − xk+1) . j,r (1 − x)2 k,r k+1,r (1 − x)3 j=0 Proof. Multiply the recurrence relation (1.10) by xj and sum over j to obtain

k k k k X j X j X j X j Pj+1,rx + Pj−1,rx − 2 Pj,rx = γ x , (3.1) j=0 j=0 j=0 j=0 in which, k k X 1 X P xj = xkP + P xj , (3.2) j+1,r k+1,r x j,r j=0 j=0

k k X j k+1 X j Pj−1,rx = γ − 1 − x Pk,r + x Pj,rx , (3.3) j=0 j=0 and k X 1 − xk+1 xj = . (3.4) 1 − x j=0 Use of (3.2), (3.3) and (3.4) in (3.1) gives the stated identity. For the triangular numbers, we have

k X xk+1 x T xj = (xT − T ) + (1 − xk+1) . (3.5) j (1 − x)2 k k+1 (1 − x)3 j=0 Corollary 8. If k is an integer, then,

k X γkTk+1 γkTk+1 P = T P − T P + = −(γ − 1)T + , (3.6) j,r k+1 k,r k k+1,r 3 k 3 j=0  k (Pk,r + Pk+1,r − 1)/4, if k is even X j  (−1) Pj,r = (3.7) j=0  {γ − (1 + Pk,r + Pk+1,r)}/4, if k is odd . In particular, k X k(k + 1)(k + 2) 1 T = = kT , (3.8) j 6 3 k+1 j=0  k k(k + 2)/4, If k is even; X j  (−) Tj = (3.9) j=0  −(k + 1)2/4, If k is odd . Corollary 9. We have ∞ X (1 − γ)x γx P xj = + . j,r (1 − x)2 (1 − x)3 j=0

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In particular, ∞ X x T xj = . (3.10) j (1 − x)3 j=0

th Lemma 1 (Partial sum of an n order sequence). Let (Xj) be any arbitrary sequence,

where Xj, j ∈ Z, satisﬁes a n + 1 term recurrence relation Xj = f1Xj−c1 + f2Xj−c2 + Pn ··· + fnXj−cn = m=1 fmXj−cm , where f1, f2, ..., fn are arbitrary non-vanishing complex functions, not dependent on j, and c1, c2, ..., cn are ﬁxed integers. Then, the following summation identity holds for arbitrary x and non-negative integer k : n o k Pn cm Pcm −j Pk j m=1 x fm j=1 x X−j − j=k−c +1 x Xj X j m Xjx = n . 1 − P xcm f j=0 m=1 m

Theorem 10. If k is an integer, then,

k X x2(γ − 1) + x {(1 − x)3 + x3} P xj = − P xk+1 j,r (1 − x)3 (1 − x)3 k+1,r j=0 (1 − 3x) P xk+3 − P xk+2 − k+3,r . (1 − x)3 k+2,r (1 − x)3

Proof. Arrange recurrence relation (1.11) as

Pj,r = Pj+3,r − 3Pj+2,r + 3Pj+1,r .

In Lemma 1 choose (f1, f2, f3) = (1, −3, 3), (c1, c2, c3) = (−3, −2, −1) and (Xj) = (Pj,r). Theorem 11. Let m and k be integers. Then

k X m(1 − x) + 2x P xj = − xk(xP − P ) + 1 − γ j+m,r (m(1 − x) − 2)(1 − x)2 k,r k+1,r j=0 γ(m(1 − x) + 2x) − (1 − xk+1) (m(1 − x) − 2)(1 − x)3 m + P − xk(P + xP ) . m(1 − x) − 2 m−1,r k+1,r k+m,r

Proof. Arrange identity (2.4) as

Tm−2Pj+m,r − Tm−1Pj+m−1,r = Tm−2Pj,r − Tm−1Pj+1,r ,

multiply through by xj and sum over j, obtaining,

k k X j X j k+1 x (Tm−2 − xTm−1) Pj+m,rx = (xTm−2 − Tm−1) Pj,rx − Tm−1x Pk+1,r j=0 j=0 k+2 + xTm−1Pm−1,r − x Tm−1Pk+m,r .

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Corollary 12. Let k and m be integers. Then,

k X γkTk+1 m P = −(γ − 1)T + + (P + P − P ) , (3.11) j+m,r k 3 2 k+m,r k+1,r m−1,r j=0

 (Pk,r+Pk+1,r−1) m k  4 + 2(m−1) (Pk+m,r − Pk+1,r + Pm−1,r), if k is even ; X j  (−1) Pj+m,r = j=0  {γ−(1+Pk,r+Pk+1,r)} m  4 − 2(m−1) (Pk+m,r − Pk+1,r − Pm−1,r), if k is odd . (3.12)

3.2 Partial sum of polygonal numbers with indices in arithmetic progression Theorem 13. Let m, n and k be integers. Then,

k ( k k ) X 3x − 1 X 1 − x (Tk+1 + xTk−1) P xj = γm2 T xj + n+jm,r 3x − x2 j 3 − x j=0 j=0 x(1 − xk) kxk+1 1 − xk+1 + (P + γmn) − + P . m,r (1 − x)2 1 − x n,r 1 − x

Proof. Diﬀerentiating both sides of the geometric sum (3.4) with respect to x gives

k X kxk+1 x(1 − xk) jxj = − + . (3.13) (1 − x) (1 − x)2 j=0

Multiply through identity (1.20) by xj and sum over j, making use of identities (3.4), (3.13) and Theorem 11 with γ = 1 (triangular number) and m = −1. Note that identity (3.13) could also be obtained directly from Theorem 7 with r = 2 (γ = 0), since Pj,2 = j. Taking limit of the identity of Theorem 13 as x approaches unity gives the sum of the polygonal numbers with indices in arithmetic progression.

Corollary 14. Let n, m and k be integers. Then,

k X γm2 P = (k − 1) + P + γmn T + (k + 1)P . mj+n,r 3 m,r k n,r j=0

In particular, for the triangular numbers we have

k X m2 T = (k − 1) + T + mn T + (k + 1)T . (3.14) mj+n 3 m k n j=0

Using x = −1 in Theorem 13 gives the alternating sum of the polygonal numbers with indices in arithmetic progression.

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Corollary 15. Let n, m and k be integers. Then,

k γmk mk k  Pn,r + 2 Pm,r + 2 n + 2 , if k is even ; X j  (−1) Pmj+n,r = j=0  k+1 − 4 {2Pm,r + γm(m(k − 1) + 2n)} , if k is odd . Dropping terms proportional to xk in Theorem 13 gives the generating function of polygonal numbers with indices in arithmetic progression.

Corollary 16. Let m and n be integers. Then,

∞ 2 2 X Pn,r (Pm,r + γmn)x γm x P xj = + + , x 6= 1 . mj+n,r 1 − x (1 − x)2 (1 − x)3 j=0

In particular, we have

∞ 2 2 X Tn (Tm + mn)x m x T xj = + + , (3.15) mj+n 1 − x (1 − x)2 (1 − x)3 j=0

∞ X x γx2 P xj = + , (3.16) j,r (1 − x)2 (1 − x)3 j=0 ∞ X x x2 T xj = + . (3.17) j (1 − x)2 (1 − x)3 j=0

4 Quadratic properties: squares and products of polygonal numbers

Here various identities and relations involving products of polygonal numbers will be devel- Pk 2 j Pk 2 j Pk j oped. These will include partial sums j=0 Pj,rx , j=0 Pj+m,rx , j=0 Pj+a,rPj+bx and the corresponding generating functions.

Lemma 2. If j is an integer, then,

2 2 2 2 4Pj,rPj−1,r = Pj,r + 4Pj−1,r − Pj−2,r + 2γPj−2,r − γ , (4.1)

2 2 2 2 4Pj,rPj−1,r = 4Pj,r + Pj−1,r − Pj+1,r + 2γPj+1,r − γ . (4.2) Proof. Square both sides of identity (1.9) and re-arrange to obtain identity (4.1). Square both sides of identity (1.10) and re-arrange to obtain identity (4.2).

Lemma 3. If j is an integer, then,

2 2 2 2 Pj+1,r − 2Pj,r + Pj−1,r = 6γPj,r + (γ − 1) + 1 .

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Proof. Square both sides of the recurrence relation

Pj+1,r + Pj−1,r = 2Pj,r + γ (4.3)

and re-arrange to obtain

2 2 2 2 2Pj+1,rPj−1,r = 4Pj,r − Pj+1,r − Pj−1,r + 4γPj,r + γ . (4.4)

The exact formula (1.2) gives

Pj+1,r − Pj−1,r = 2 + γ(2j − 1) . (4.5)

Squaring both sides of each of (4.3) and (4.5) and subtracting, we ﬁnd

2 Pj+1,rPj−1,r = Pj,r − γPj,r + γ − 1 . (4.6)

Eliminating Pj+1,rPj−1,r between (4.4) and (4.6) yields the identity stated in Lemma 3. Lemma 4. Let a, b and j be integers. Then,

1 2 1 2 1 2 Pj+a,rPj+b,r = Pj+a,r + Pj+b,r − Pa−b,r 2 2 2 (4.7) γ2 − (a − b)2(j + b)2 − γ(a − b)P (j + b) , 2 a−b,r

1 2 1 2 1 2 1 2 Pj+a+1,rPj+b+1,r − Pj+a,rPj+b,r = Pj+a+1,r + Pj+b+1,r − Pj+a,r − Pj+b,r 2 2 2 2 (4.8) γ2 − (a − b)2(2j + 2b + 1) − γ(a − b)P , 2 a−b,r

1 a − b + 1 P P − P P = P 2 − P 2 j+a+1,r j+b+1,r j+a,r j+b,r 2 a − b − 1 j+a,r j+b+1,r 1 a − b − 1 + P 2 − P 2 , |a − b|= 6 1 . 2 a − b + 1 j+b,r j+a+1,r (4.9)

Proof. Set j = 1 in (1.20) and write j for n:

Pj+m,r − Pj,r = Pm,r + γmj .

squaring both sides and re-arranging gives

2 2 2 2 2 2 2Pj+m,rPj,r = Pj+m,r + Pj,r − Pm,r − 2γmjPm,r − γ m j .

Set j → j + b, m → a − b to obtain identity (4.7). Shift the index in (4.7) and subtract identity (4.7) from the result. This gives identity (4.8). Arrange identity (2.4) as

Tm−2Pj+m,r + Tm−1Pj+1,r = Tm−2Pj,r + Tm−1Pj+m−1,r ,

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and set j → j + a and m = b − a + 1 to obtain

Tb−a−1Pj+b+1,r + Tb−aPj+a+1,r = Tb−a−1Pj+a,r + Tb−aPj+b,r .

Squaring both sides of the above relation and rearranging the terms we get

1 Ta−b 2 2 Pj+a+1,rPj+b+1,r − Pj+a,rPj+b,r = Pj+a,r − Pj+b+1,r 2 Ta−b−1 1 Ta−b−1 2 2 + Pj+b,r − Pj+a+1,r ; 2 Ta−b from which identity (4.9) follows.

Theorem 17. If j and m are integers, then,

2 2 2 2 2 2 Tm−3(Pj+m+1,r − Pj,r) − (m − 3)(m + 1)(Pj+m,r − Pj+1,r) + Tm(Pj+m−1,r − Pj+2,r) = 0 .

Proof. Write identity (2.4) as

Tm−2(Pj+m,r − Pj) = Tm−1(Pj+m−1 − Pj+1) .

Square both sides of the above relation and re-arrange the terms, obtaining,

2 2 2Tm−2Pj,rPj+m,r − 2Tm−1Pj+1,rPj+m−1,r 2 2 2 2 2 2 (4.10) = Tm−2(Pj+m,r + Pj,r) − Tm−1(Pj+m−1,r + Pj+1,r) .

Write j + 1 for j in relation (4.10) to get

2 2 2Tm−2Pj+1,rPj+m+1,r − 2Tm−1Pj+2,rPj+m,r 2 2 2 2 2 2 (4.11) = Tm−2(Pj+m+1,r + Pj+1,r) − Tm−1(Pj+m,r + Pj+2,r) .

Subtract (4.10) from (4.11) to get

2 2Tm−2(Pj+1,rPj+m+1,r − Pj,rPj+m,r) 2 − 2Tm−1(Pj+2,rPj+m,r − Pj+1,rPj+m−1,r) 2 2 2 2 2 (4.12) = Tm−2(Pj+m+1,r + Pj+1,r − Pj+m,r − Pj,r) 2 2 2 2 2 − Tm−1(Pj+m,r + Pj+2,r − Pj+m−1,r − Pj+1,r) .

Using Lemma 4, identity (4.12) is 2 Tm−1 2 2 Tm 2 2 Tm−2 (Pj,r − Pj+m+1,r) + (Pj+m,r − Pj+1,r) Tm Tm−1 2 Tm−2 2 2 Tm−3 2 2 − Tm−1 (Pj+m−1,r − Pj+2,r) + (Pj+1,r − Pj+m,r) Tm−3 Tm−2 2 2 2 2 2 = Tm−2(Pj+m+1,r + Pj+1,r − Pj+m,r − Pj,r) 2 2 2 2 2 − Tm−1(Pj+m,r + Pj+2,r − Pj+m−1,r − Pj+1,r) . The identity given in the theorem follows upon clearing fractions and simpliﬁcation.

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Corollary 18. If j is an integer, then,

2 2 2 2 2 2 Pj,r − 4Pj+1,r + 5Pj+2,r − 5Pj+4,r + 4Pj+5,r − Pj+6,r = 0 , (4.13)

2 2 2 2 2 2 Pj,r − 5Pj+1,r + 10Pj+2,r − 10Pj+3,r + 5Pj+4,r − Pj+5,r = 0 , (4.14)

2 2 2 2 2 2 2Pj,r − 7Pj+1,r + 7Pj+2,r − 7Pj+5,r + 7Pj+6,r − 2Pj+7,r = 0 . (4.15) Proof. To prove (4.13) set m = −3 or m = 5 in Theorem 17. Setting m = 4 or m = −2 in Theorem 17 proves (4.14) while (4.15) follows from m = −4 in the theorem. Theorem 19. Let m and j be integers. Then,

2 2 2 2 m(m − 2)Pj+m+1,r − 2(m − 4)Pj+m,r + m(m + 2)Pj+m−1,r 2 2 2 = m(m + 2)Pj+1,r − 2(m − 4)Pj,r 2 + m(m − 2)(Pj−1,r + 2γPj+m+1,r − 2γPj−1,r) . Proof. Arrange identity (2.4) as

Tm−2Pj+m,r − Tm−1Pj+m−1,r = Tm−2Pj,r − Tm−1Pj+1,r .

Square both sides of the above identity and use identities (4.1) and (4.2) to eliminate the products Pj+m,rPj+m−1,r and Pj,rPj+1,r. The result is

2 2 2 2 2 Tm−1Tm−2Pj+m+1,r + (2Tm−2 − 4Tm−1Tm−2)Pj+m,r + (2Tm−1 − Tm−1Tm−2)Pj+m−1,r 2 2 2 2 2 = (2Tm−1 − Tm−1Tm−2)Pj+1,r + (2Tm−2 − 4Tm−1Tm−2)Pj,r + Tm−1Tm−2Pj−1,r (4.16)

+ 2γTm−1Tm−2(Pj+m+1,r − Pj−1,r) , from which the identity of the theorem follows upon substituting for the triangular numbers.

We have the following examples:

2 2 2 2 Pj+3,r − 3Pj+2,r + 3Pj+1,r − Pj,r = 2γ(Pj+3,r − Pj,r) , (4.17)

2 2 2 2 3Pj+2,r − 3Pj+1,r + Pj,r − Pj+3,r = 6γ(Pj+1,r − Pj+2,r) . (4.18) Identities (4.17) and (4.18) come from evaluating the identity stated in Theorem 19 at m = 1 and m = −1, respectively.

4.1 Partial sums and generating functions Theorem 20. If k is an integer, then,

k 2 k k+1 X 6γx x γx (1 − x ) x (Pk+1,r − xPk,r) P 2 xj = + − j,r (1 − x)2 (1 − x)2 (1 − x)3 (1 − x)2 j=0 ((γ − 1)2 + 1)x(1 − xk+1) xk+1(P 2 − xP 2 ) (γ − 1)2x + − k+1,r k,r − . (1 − x)3 (1 − x)2 (1 − x)2

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Proof. Multiply through the identity of Lemma 3 and sum over j to obtain

k k k X 2 j X 2 j X 2 j Pj+1,rx − 2 Pj,rx + Pj−1,rx j=0 j=0 j=0 k k X j 2 X j = 6γ Pj,rx + ((γ − 1) + 1) x ; j=0 j=0 which, by shifting the index in each sum appropriately and re-arranging can be written

k k k 2 X 2 j X j 2 X j (1 − x) Pj,rx = 6γx Pj,rx + ((γ − 1) + 1)x x j=0 j=0 j=0 k+1 2 k+2 2 2 − x Pk+1,r + x Pk,r − (γ − 1) x ; from which the result follows when we use Theorem 7. Corollary 21. If k is an integer, then,

k X γ 1 1 P 2 = {(k + 3) + P − P }γT + kP 2 − (2k − 3)P 2 T j,r 5 k,r k+1,r k−1 3 k−1,r 3 k,r k+1 j=0 (4.19) 1 + (k + 1)T P 2 , 3 k−1 k+1,r

γ 1 2 2 2 k  8 (γ − {3(Pk+1,r − Pk,r) − 1}) + 8 Pk+1,r − Pk−1,r + 4Pk,r , if k is even ; X j 2  (−1) Pj,r = γ 1 j=0  2 2 2 8 (−2γ + {3(Pk+1,r − Pk,r) + 1}) − 8 Pk+1,r − Pk−1,r + 4Pk,r , if k is odd . (4.20) Proof. Repeated application of L’Hospital’s rule to the identity of Theorem 20 at x = 1 yields identity (4.19); while (4.20) comes from evaluating the identity at x = −1. Corollary 22. We have ∞ X 6γx x γx2 {(γ − 1)2 + 1}x (γ − 1)2x P 2 xj = + + − . j,r (1 − x)2 (1 − x)2 (1 − x)3 (1 − x)3 (1 − x)2 j=0 For the triangular numbers, γ = 1 and we have the generating function of the squares of triangular numbers to be given by ∞ X 6x2 x T 2xj = + . (4.21) j (1 − x)5 (1 − x)3 j=0 Theorem 23. If k is an integer, then,

k X (γ − 1)2x4 − {1 + 2γ(1 − 2γ)} x3 + {γ(γ + 4) − 1} x2 + x P 2 xj = j,r (1 − x)5 j=0 ((1 − x)5 + x5) ((1 − x)5 + x5 − 5x4) − P 2 xk+1 − P 2 xk+2 (1 − x)5 k+1,r (1 − x)5 k+2,r (1 − 5x + 10x2) (1 − 5x) P 2 xk+5 − P 2 xk+3 − P 2 xk+4 − k+5,r . (1 − x)5 k+3,r (1 − x)5 k+4,r (1 − x)5

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Proof. Write recurrence relation (4.14) as

2 2 2 2 2 2 Pj,r = 5Pj+1,r − 10Pj+2,r + 10Pj+3,r − 5Pj+4,r + Pj+5,r .

In Lemma 1 choose (f1, f2, f3, f4, f5) = (5, −10, 10, −5, 1), (c1, c2, c3, c4, c5) = (−1, −2, −3, −4, −5) 2 and (Xj) = (Pj,r). Corollary 24. We have

∞ X (γ − 1)2x4 − {1 + 2γ(1 − 2γ)} x3 + {γ(γ + 4) − 1} x2 + x P 2 xj = . j,r (1 − x)5 j=0

Theorem 25. Let m and k be integers. Then,

k k X m(m − 2)x2 − 2(m2 − 4)x + m(m + 2) X P 2 xj = P 2 xj j+m,r m(m + 2)x2 − 2(m2 − 4)x + m(m − 2) j,r j=0 j=0 2 Pk j Pk j 2γm(m − 2) x j=0 Pj,rx − j=0 Pj+m,rx − m(m + 2)x2 − 2(m2 − 4)x + m(m − 2) 2γm(m − 2) xk+2P + xk+1P − xP − P + k,r k+m+1,r −1,r m,r m(m + 2)x2 − 2(m2 − 4)x + m(m − 2) m(m + 2)(xk+2P 2 + P 2 xk+1 − xP 2 ) + k+m,r k+1,r m−1,r m(m + 2)x2 − 2(m2 − 4)x + m(m − 2) m(m − 2)(xk+2P 2 + xk+1P 2 − xP 2 − P 2 ) − k,r k+m+1,r −1,r m,r . m(m + 2)x2 − 2(m2 − 4)x + m(m − 2)

Proof. Multiply through the identity of Theorem 19 by xj and sum over j, shifting the index in each sum as appropriate.

Corollary 26. Let m and k be integers. Then,

k X m(m − 2) P 2 = (P − γ)2 − (P − γ)2 − (P − γ)2 + 1 j+m,r 8 m,r k+m+1,r k,r j=0 γm2(m − 2) m(m + 2) + {P + P − P } + P 2 + P 2 − P 2 8 k+m,r k+1,r m−1,r 8 k+m,r k+1,r m−1,r γ 1 1 + {(k + 3) + P − P }γT + kP 2 − (2k − 3)P 2 T 5 k,r k+1,r k−1 3 k−1,r 3 k,r k+1 1 + (k + 1)T P 2 . 3 k−1 k+1,r (4.22)

Proof. Set x = 1 in Theorem 25 making use of (3.6), (3.11) and (4.19).

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Theorem 27. Let m and k be integers. Then,

k 2 k X x Tm−3 − (m − 3)(m + 1)x + Tm X P 2 xj = P 2 xj j+m,r xT − (m − 3)(m + 1)x2 + T x3 j,r j=0 m−3 m j=0 k+1 2 k 2 Tmx Pk+2,r + (Tm − (m − 3)(m + 1)x)x Pk+1,r + 2 Tm−3 − (m − 3)(m + 1)x + Tmx k+2 2 k+1 2 Tmx Pk+m,r − Tm−3x Pk+m+1,r + 2 Tm−3 − (m − 3)(m + 1)x + Tmx 2 2 Tm + xTmPm−1,r − Tm−3Pm,r − 2 2 . Tm−3 − (m − 3)(m + 1)x + Tmx

Proof. Multiply through the identity of Theorem 17 by xj and sum over j, shifting the index in each sum as appropriate.

Theorem 28. Let a, b and k be integers such that |a − b > 1. Let x ∈ C, x 6= 1. Then,

k k k X (T 2 − xT 2 ) X (T 2 − xT 2 ) X 2 P P xj = − b−a b−a−1 P 2 xj − b−a−1 b−a P 2 xj j+a j+b T T (1 − x) j+a T T (1 − x) j+b j=0 b−a b−a−1 j=0 b−a b−a−1 j=0 T (P 2 − xk+1P 2 ) T (P 2 − xk+1P 2 ) + b−a a k+a+1 + b−a−1 b k+b+1 Tb−a−1 (1 − x) Tb−a (1 − x) xk+1 P P − 2 P P + 2 a b . (1 − x) k+a+1 k+b+1 (1 − x)

Proof. Multiply through identity (4.9) by xj and sum over j, shifting the index in each sum as appropriate.

Corollary 29. Let a, b and k be integers. Then,

k X a(a − 2) 2 P P = (P − γ)2 − (P − γ)2 − (P − γ)2 + 1 j+a,r j+b,r 8 a,r k+a+1,r k,r j=0 b(b − 2) + (P − γ)2 − (P − γ)2 − (P − γ)2 + 1 8 b,r k+b+1,r k,r a(a + 2) b(b + 2) + P 2 + P 2 − P 2 + P 2 + P 2 − P 2 8 k+a,r k+1,r a−1,r 8 k+b,r k+1,r b−1,r γa2(a − 2) γb2(b − 2) + {P + P − P } + {P + P − P } 8 k+a,r k+1,r a−1,r 8 k+b,r k+1,r b−1,r γ 1 1 + 2T {(k + 3) + P − P }γT + kP 2 − (2k − 3)P 2 k+1 5 k,r k+1,r k−1 3 k−1,r 3 k,r 2 + (k + 1) T P 2 − P {P + γ(a − b)(k + 2b)} 3 k−1 k+1,r a−b,r a−b,r γ2(a − b)2 − (k + 1) (2k2 + k + 6kb + 6b2) . 6 Proof. Sum each term in identity (4.7) from 0 to k, making use of Corollary 26.

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5 Powers of polygonal numbers

We have seen that, identity (1.11),

Pj,r − 3Pj+1,r + 3Pj+2,r − Pj+3,r = 0 ,

and that, identity (4.14),

2 2 2 2 2 2 Pj,r − 5Pj+1,r + 10Pj+2,r − 10Pj+3,r + 5Pj+4,r − Pj+5,r = 0 . More generally, on account of Euler’s ﬁnite diﬀerence theorem, Gould [6, identity (10.1)],  t 0, if 0 ≤ m < t ; X t  (−1)t+s sm = (5.1) s s=0  (−1)tt!, if m = t ,

we have the result stated in the next theorem. Theorem 30. Let n be a non-negative integer. Then,

2n+1 X 2n + 1 (−1)s P n = 0 . s j+s,r s=0 Here are a few more explicit examples:

3 3 3 3 3 Pj,r − 7Pj+1,r + 21Pj+2,r − 35Pj+3,r + 53Pj+4,r 3 3 3 (5.2) − 21Pj+5,r + 7Pj+6,r − Pj+7 = 0 ,

4 4 4 4 4 4 Pj,r − 9Pj+1,r + 36Pj+2,r − 84Pj+3,r + 126Pj+4,r − 126Pj+5,r 4 4 4 4 (5.3) + 84Pj+6,r − 36Pj+7,r + 9Pj+8,r − Pj+9,r = 0 ,

5 5 5 5 5 5 Pj,r − 11Pj+1,r + 55Pj+2,r − 165Pj+3,r + 330Pj+4,r − 462Pj+5,r 5 5 5 5 5 5 (5.4) + 462Pj+6,r − 330Pj+7,r + 165Pj+8,r − 55Pj+9,r + 11Pj+10,r − Pj+11,r = 0 .

5.1 Partial sums and generating functions Theorem 31. Let n be a non-negative integer and k any integer. Then,

k 2n+1 ( k+s ) X 1 X 2n + 1 X P n xj = (−1)s x2n−s+1 P n xj j,r (1 − x)2n+1 s j,r j=0 s=0 j=k+1 2n+1 ( s−1 ) 1 X 2n + 1 X − (−1)s x2n−s+1 P n xj . (1 − x)2n+1 s j,r s=0 j=0

Proof. Multiply through the identity of Theorem 30 by xj and sum,

2n+1 ( k ) X 2n + 1 X (−1)s P n xj = 0 . s j+s,r s=0 j=0

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Using (1.30), this is

2n+1 ( k s−1 k+s !) X 2n + 1 X X X (−1)s x−s P n xj − x−s P n xj + x−s P n xj = 0 . s j,r j,r j+s,r s=0 j=0 j=0 j=k+1 Clearing brackets and multiplying through by x2n+1, we have

k 2n+1 2n+1 s−1 X X 2n + 1 X 2n + 1 X P n xj (−1)s x2n−s+1 − (−1)s x2n−s+1 P n xj j,r s s j,r j=0 s=0 s=0 j=0 2n+1 k+s X 2n + 1 X + (−1)s x2n−s+1 P n xj = 0 . s j,r s=0 j=k+1

Pk n j The result now follows upon making j=0 Pj,rx subject in the above equation. Note that

2n+1 X 2n + 1 (−1)s x2n−s+1 = −(1 − x)2n+1 . s s=0

Observe that Theorem 10 and Theorem 23 are special cases of Theorem 31. Taking limit as k approaches inﬁnity in Theorem 31, we obtain the generating function of the powers of polygonal numbers as stated in the next result. Corollary 32. Let n be a non-negative integer. Then,

∞ 2n+1 ( s−1 ) X 1 X 2n + 1 X P n xj = (−1)s−1 x2n−s+1 P n xj . j,r (1 − x)2n+1 s j,r j=0 s=1 j=0

6 Weighted sums

In this section we present some weighted ordinary and binomial summation identities involving the polygonal numbers. The general summation identities which we require for this purpose are given in Lemmata 5 and 6 and can be proved by induction (see Adegoke [1]).

6.1 Ordinary summation identities

Lemma 5. Let (Xn) be any arbitrary sequence. Let Xn, n ∈ Z, satisfy a recurrence relation Xn = f1Xn−c + f2Xn−d + λ, where f1 and f2 are non-vanishing complex functions, not dependent on n, and c and d are integers. Then,

k X λ(1 − f k+1) f f jX = X − f k+1X − 1 , (6.1) 2 1 n−d−jc n 1 n−(k+1)c 1 − f j=0 1

k X λ(1 − f k+1) f f jX = X − f k+1X − 2 , (6.2) 1 2 n−c−jd n 2 n−(k+1)d 1 − f j=0 2

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Theorem 33. Let m, n, k and r be integers. Then,

k X j k+1 2 k+1 2 Pn−2m−mj,r = 2 Pn−(k+1)m,r − Pn,r + m γ(2 − 1) , (6.3) j=0

1 1 1 k  2 2 Pn,r + 2 Pn−2m(k+1),r − 2 m γ, if k is even , X j  (−1) Pn−m−2mj,r = (6.4) 1 1 j=0  2 Pn,r − 2 Pn−2m(k+1),r, if k is odd , 6.2 Binomial summation identities

Lemma 6. Let (Xn) be any arbitrary sequence. Let Xn, n ∈ Z, satisfy a recurrence relation Xn = f1Xn−c + f2Xn−d + λ, where f1 and f2 are non-vanishing complex functions, not dependent on n, and c and d are integers. Then,

k k X k λ((f1 + f2) − 1) f jf k−jX = X − , (6.5) j 1 2 n−kd+(d−c)j n f + f − 1 j=0 1 2

k k k X k λ((1 − f2) − f ) (−1)k (−1)j f k−jX = f kX + 1 , (6.6) j 2 n+k(c−d)+dj 1 n 1 − f − f j=0 2 1 and k k k X k λ((1 − f1) − f ) (−1)k (−1)j f k−jX = f kX + 2 , (6.7) j 1 n+k(d−c)+cj 2 n 1 − f − f j=0 1 2 for k a non-negative integer. Theorem 34. Let m, n, k and r be integers. Then,

k X k (−1)k (−1)j 2jP = P − m2kγ , (6.8) j n−2mk+mj,r n,r j=0

k X k P = 2kP + 2k−1m2kγ , (6.9) j n−mk+2mj,r n,r j=0

k X k (−1)j 2k−jP = P − m2kγ . (6.10) j n+mk+mj,r n,r j=0 Theorem 35. Let m, n, k and r be integers. Then,

k X k (−1)k (−1)j (m + 1)jmk−jP = P − kγT , (6.11) j n−k(m+1)+j,r n,r m j=0

k X k mk−jP = (m + 1)kP + k(m + 1)k−1γT , (6.12) j n−k+(m+1)j,r n,r m j=0

k X k (−1)j (m + 1)k−jP = mkP − kmk−1γT . (6.13) j n+k+mj,r n,r m j=0

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6.3 Double binomial summation identities 7 Some triangular-numbers-speciﬁc properties

We will conclude this study by presenting some results which are speciﬁc to triangular numbers.

Theorem 36. Let n be an integer. Then,

TnTn−1 = TTn−1 + TTn−1−1 .

2 Proof. Since Tn − Tn−1 = n and Tn + Tn−1 = n , we have

2 2 Tn + Tn−1 = Tn + Tn−1 − 2TnTn−1 .

Thus, 2 2 2TnTn−1 = (Tn − Tn) + (Tn−1 − Tn−1) , and the result follows. The implication of Theorem 36 is that the product of any two consecutive triangular numbers is the sum of two certain triangular numbers. On account of the summation identity (1.34), the identity given in Theorem 36 invokes the next result.

Corollary 37. If k is an integer, then,

k X j k (−1) TjTj−1 = (−1) T Tk+1Tk−1 . T j=0 k

Next we give a recursive identity involving any three consecutive triangular numbers.

Theorem 38. If n is an integer, n 6= 0, n 6= −1, then,

T T 2 2 2 2 2 T T n+1 n−1 = T T T ··· T n+1 n−1 ··· . Tn Tn Tn Tn Tn Tn Tn

Here we have written T (j) ≡ Tj. Proof. Setting m = 1 in identity (1.25) gives

2 Tn+1Tn−1 = Tn − Tn = 2TTn−1 . (7.1)

But, 2 Tn+1Tn−1 Tn+1Tn−1 = Tn − Tn ⇒ = Tn − 1 . (7.2) Tn Using (7.2) in (7.1), we have

Tn+1Tn−1 = 2T Tn+1Tn−1 . Tn Thus, Tn+1Tn−1 2 = T Tn+1Tn−1 , Tn Tn Tn

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from which we get again Tn+1Tn−1 2 = T 2 T , (7.3) T T Tn Tn+1Tn−1 n n Tn

Tn+1Tn−1 2 2 = T T 2 . (7.4) Tn T Tn Tn Tn Tn+1Tn−1 Tn In short, Tn+1Tn−1 2 2 = T T 2 . Tn ... Tn Tn Tn

References

[1] Kunle Adegoke, Weighted sums of some second-order sequences, The Fibonacci Quar- terly 56:3 (2018), 252–262.

[2] C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, The Fibonacci Quarterly 52:4 (2014), 336–343.

[3] E. Deza and M. M. Deza, Figurate Numbers, World Scientiﬁc, 2012.

[4] L. E. Dickson, History of the Theory of Numbers, Volume II: Diophantine Analysis, Dover Publications, 2005.

[5] A. S. Garge and S. A. Shirali, Triangular numbers, Resonance 17:7 (2012), 672–681.

[6] H. G. Gould, Combinatorial Identities: Table I: Intermediate Techniques for Summing Finite Series, http://www.math.wvu.edu/~gould/Vol.4.PDF.

[7] V. E. Hoggatt, JR., and M. Bicknell, Triangular numbers, The Fibonacci Quarterly 12:3 (1974), 221–230.