Inder J. Taneja Single Letter Patterned Representations and Fibonacci Sequence Values, https://inderjtaneja.com Zenodo, Feb. 06, 2019, pp. 1-40, https://doi.org/10.5281/zenodo.2558522 Single Letter Patterned Representations and Fibonacci Sequence Values1 Inder J. Taneja2 Abstract This work brings representations of palindromic and number patterns in terms of single letter ”a”. Some examples of prime patterns are also considered. Different classifications of palindromic patterns are considered, such as palindromic decomposition, double symmetric patterns, number patterns decompositions, etc. Number patterns with powers are also studied. Some extensions to Pythagorean triples are also given. Study towards Fibonacci sequence and its extensions are also made. This work is revised and enlarged version of author’s previous work done in 2015. Contents 1 Single Letter Representations 1 2 Representations of Palindromic and Number Patterns 2 2.1 Number Patterns with Palindromic Decompositions . .2 2.2 Palindromic Patterns with Number Pattern Decompositions . .7 2.3 Palindromic Patterns with Palindromic Decompositions . 14 2.4 Squared Number Patterns . 18 2.5 Cyclic-Type Patterns . 21 2.6 Doubly Symmetric Patterns . 27 2.7 Number Patterns with Number Pattern Decompositions . 30 2.8 Semi-Prime Patterns . 34 2.9 Fibonacci Sequences and Extensions . 35 1 Single Letter Representations Let us consider f n(10) = 10n + 10n−1 + ... + 102 + 10 + 100, For a 2 f1, 2, 3, 4, 5, 6, 7, 8, 9g, we can write a f n(10) = aaa...a , | {z } (n+1)−times 1It is revised and enlarged version of authors previous work [11, 12] appeared in 2015 2Formerly, Professor of Mathematics, Federal University of Santa Catarina, Florian´opolis,SC, Brazil (1978-2012). Also worked at Delhi University, India (1976-1978). E-mail: [email protected]; Web-site: https://inderjtaneja.com; Twitter: @IJTANEJA; Instagram: @crazynumbers. 1 Inder J. Taneja Single Letter Patterned Representations and Fibonacci Sequence Values, https://inderjtaneja.com Zenodo, Feb. 06, 2019, pp. 1-40, https://doi.org/10.5281/zenodo.2558522 In particular, aa aa = f 1(10) = a10 + a =) 11 := a aaa aaa = f 2(10) = a102 + a10 + a =) 111 := a aaaa aaaa = f 3(10) = a103 + a102 + a10 + a =) 1111 := a aaaaa aaaaa = f 4(10) = a104 + a103 + a102 + a10 + a =) 11111 := a ...... In [12, 14] author wrote natural numbers in terms of single letter ”a”. See some examples below: aa − a 5 := a + a aaa + a 56 := a + a aaaaa + aaaa 582 := aa + aa − a aaaa + aaa + aa 1233 := a aaaaa − aaaa − aaa + aa 4950 := . a + a for any 2 f1, 2, 3, 4, 5, 6, 7, 8, 9g. The work on representations of natural numbers using single letter is first of its kind [20, 21]. Author also worked [16, 17, 18, 19], with representation of numbers using single digit for each value from 1 to 9 sepa- rately. Representation of numbers using all the digits from 1 to 9 in increasing and decreasing orders is done by author [7, 8, 9]. Comments to this work can be seen at author’s site [10]. Different studies on numbers, such as, selfie numbers, running expressions, etc. refer to author’s work web-site: https://inderjtaneja.com. Our aim is to write number patterns using single letter ”a”. For studies on patterns refer to [1, 2, 4]. This work revised and enlarged version of authors previous work [11, 12] done in 2015. 2 Representations of Palindromic and Number Patterns Below are examples of palindromic and number patterns in terms of single letter ”a”. These examples are divided in subsections. All the examples are followed by their respective decompositions. The following patterns are considered: 2.1 Number Patterns with Palindromic Decompositions Here below are examples of palindromic patterns with palindromic decompositions. Example 1. The following example is represented in two different forms. One with product decomposition and another with potentiation. 2 Inder J. Taneja Single Letter Patterned Representations and Fibonacci Sequence Values, https://inderjtaneja.com Zenodo, Feb. 06, 2019, pp. 1-40, https://doi.org/10.5281/zenodo.2558522 (i) Product aa × aa 121 = 11 × 11 := a × a aaa × aaa 12321 = 111 × 111 := a × a aaaa × aaaa 1234321 = 1111 × 1111 := a × a aaaaa × aaaaa 123454321 = 11111 × 11111 := a × a aaaaaa × aaaaaa 12345654321 = 111111 × 111111 := a × a aaaaaaa × aaaaaaa 1234567654321 = 1111111 × 1111111 := a × a aaaaaaaa × aaaaaaaa 123456787654321 = 11111111 × 11111111 := a × a aaaaaaaaa × aaaaaaaaa 12345678987654321 = 111111111 × 111111111 := a × a Since 11 × 11 = 112, 111 × 111 = 1112, etc. Below is an alternative way of writing the same pattern: (ii) Potentiation a+a aa a 121 = 112 := a a+a aaa a 12321 = 1112 := a a+a aaaa a 1234321 = 11112 := a a+a aaaaa a 123454321 = 111112 := a a+a aaaaaa a 12345654321 = 1111112 := a a+a aaaaaa a 1234567654321 = 11111112 := a a+a aaaaaaaa a 123456787654321 = 111111112 := a a+a aaaaaaaaa a 12345678987654321 = 1111111112 := . a 3 Inder J. Taneja Single Letter Patterned Representations and Fibonacci Sequence Values, https://inderjtaneja.com Zenodo, Feb. 06, 2019, pp. 1-40, https://doi.org/10.5281/zenodo.2558522 Example 2. a × aa 11 = 1 × 11 := a × a aa × aaa 1221 = 11 × 111 := a × a aaa × aaaa 123321 = 111 × 1111 := a × a aaaa × aaaaa 12344321 = 1111 × 11111 := a × a aaaaa × aaaaaa 1234554321 = 11111 × 111111 := a × a aaaaaa × aaaaaaa 123456654321 = 111111 × 1111111 := a × a aaaaaaa × aaaaaaaa 12345677654321 = 1111111 × 11111111 := a × a aaaaaaaa × aaaaaaaaa 1234567887654321 = 11111111 × 111111111 := a × a aaaaaaaaa × aaaaaaaaaa 123456789987654321 = 111111111 × 1111111111 := a × a • Palindromic-Type Pythagorean Triples Similar to Example 1, there are lot of palindromic-type patterns with Pythagorean triples. See below the two examples, Example 3. 0992 + 202 = 1 012 12 0992 + 2202 = 121 012 1232 0992 + 22202 = 12321 012 123432 0992 + 222202 = 1234321 012 12345432 0992 + 2222202 = 123454321 012 1234565432 0992 + 22222202 = 12345654321 012 123456765432 0992 + 222222202 = 1234567654321 012 12345678765432 0992 + 2222222202 = 123456787654321 012 1234567898765432 0992 + 22222222202 = 12345678987654321 012 It is based on the pythagorean triple (99, 20, 101). 4 Inder J. Taneja Single Letter Patterned Representations and Fibonacci Sequence Values, https://inderjtaneja.com Zenodo, Feb. 06, 2019, pp. 1-40, https://doi.org/10.5281/zenodo.2558522 Example 4. 0192 + 1802 = 1 812 12 0192 + 19802 = 121 812 1232 0192 + 199802 = 12321 812 123432 0192 + 1999802 = 1234321 812 12345432 0192 + 19999802 = 123454321 812 1234565432 0192 + 199999802 = 12345654321 812 123456765432 0192 + 1999999802 = 1234567654321 812 12345678765432 0192 + 19999999802 = 123456787654321 812 1234567898765432 0192 + 199999999802 = 12345678987654321 812 It is based on the pythagorean triple (19, 180, 181). More study on the Examples 3 and 4 can be seen in author’s [22, 23]. Example 5. The following example is represented in two different forms. One with product decomposition and another with potentiation. (i) Product aa × (aaa − aa − a) 1089 = 11 × 99 := a × a aaa × (aaaa − aaa − a) 110889 = 111 × 999 := a × a aaaa × (aaaaa − aaaa − a) 11108889 = 1111 × 9999 := a × a aaaaa × (aaaaaa − aaaaa − a) 1111088889 = 11111 × 99999 := a × a aaaaaa × (aaaaaaa − aaaaaa − a) 111110888889 = 111111 × 999999 := . a × a The above decomposition can also we written as 11 × 99 = 332, 111 × 999 = 3332, etc. We can rewrite the above representation using as potentiation: (ii) Potentiation ( a+a ) aa + aa + aa a 1089 = 11 × 99 = 332 := a ( a+a ) aaa + aaa + aaa a 110889 = 111 × 999 = 3332 := a ( a+a ) aaaa + aaaa + aaaa a 11108889 = 1111 × 9999 = 33332 := a ( a+a ) aaaaa + aaaaa + aaaaa a 1111088889 = 11111 × 99999 = 333332 := a ( a+a ) aaaaaa + aaaaaa + aaaaaa a 111110888889 = 111111 × 999999 = 3333332 := a 5 Inder J. Taneja Single Letter Patterned Representations and Fibonacci Sequence Values, https://inderjtaneja.com Zenodo, Feb. 06, 2019, pp. 1-40, https://doi.org/10.5281/zenodo.2558522 The reverse of 1089 is 9801 has following interesting pattern: Example 6. The following example is represented in two different forms. One with product decomposition and another with potentiation. (i) Potentiation a+a aaa − aa − a a 9801 = 992 := a × a a+a aaaa − aa − a a 998001 = 9992 := a × a a+a aaaaa − aa − a a 99980001 = 99992 := a × a a+a aaaaaa − aa − a a 9999800001 = 999992 := a × a a+a aaaaaaa − aa − a a 999998000001 = 9999992 := a × a We can rewrite the above representation in a direct way without potentiation: (ii) Without Potentiation (aaaa − aa − aa) × (aa − a − a) 9801 = 992 := a × a (aaaaaa − aaa − aaa) × (aa − a − a) 998001 = 9992 := a × a (aaaaaaaa − aaaa − aaaa) × (aa − a − a) 99980001 = 99992 := a × a (aaaaaaaaaa − aaaaa − aaaaa) × (aa − a − a) 9999800001 = 999992 := a × a (aaaaaaaaaaaa − aaaaaa − aaaaaa) × (aa − a − a) 999998000001 = 9999992 := a × a More examples on potentiation are given in Section 2.4. See examples 32, 33, 34, 35 and 36. Example 7. aa × (aaa − aa − a) × (aa − a − a − a − a) 7623 = 11 × 9 × 77 := a × a × a aaa × (aaaa − aaa − a) × (aa − a − a − a − a) 776223 = 111 × 9 × 777 := a × a × a aaaa × (aaaaa − aaaa − a) × (aa − a − a − a − a) 77762223 = 1111 × 9 × 7777 := a × a × a aaaaa × (aaaaaa − aaaaa − a) × (aa − a − a − a − a) 7777622223 = 11111 × 9 × 77777 := a × a × a aaaaaa × (aaaaaaa − aaaaaa − a) × (aa − a − a − a − a) 777776222223 = 111111 × 9 × 777777 := a × a × a 6 Inder J. Taneja Single Letter Patterned Representations and Fibonacci Sequence Values, https://inderjtaneja.com Zenodo, Feb. 06, 2019, pp. 1-40, https://doi.org/10.5281/zenodo.2558522 2.2 Palindromic Patterns with Number Pattern Decompositions Here below are examples palindromic patterns decomposed in number patterns. Example 8. a 1 = 0 × 9 + 1 := a aa 11 = 1 × 9 + 2 := a aaa 111 = 12 × 9 + 3 := a aaaa 1111 = 123 × 9 + 4 := a aaaaa 11111 = 1234 × 9 + 5 := a aaaaaa 111111 = 12345 × 9 + 6 := a aaaaaaa 1111111 = 123456 × 9 + 7 := a aaaaaaaa 11111111 = 1234567 × 9 + 8 := a aaaaaaaaa 111111111 = 12345678 × 9 + 9 := a aaaaaaaaaa 1111111111 = 123456789 × 9 + 10 := a Example 9.