Figurative Numbers

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Figurative Numbers 1. Write down the first 10 square numbers. Why are they called square numbers? 2. If you build triangles out of dots and count the dots you get the triangular numbers. Here are the first two triangular numbers represented geometrically (the first is admittedly not much of a triangle): Find the first 10 triangular numbers. 3. Find a formula for the n-th triangular number. 4. What happens if you add two consecutive triangular numbers? Why does this happen? Can you prove your result geometrically? Can you prove your result algebraically? 5. Let T (n) represent the n-th triagular number. Write your previous results using this T (n) notation. (How do you write your general results for what you have discovered?) 6. If you add up triagular numbers you get the tetrahedral numbers. For example, the first several tetrahedral numbers are: 1; 3; 6;::: Find the first several tetrahedral numbers. Can you find a formula for the n-tetrahedral number? 1 Figurative Numbers Part 2 1. Write down the first 10 pentagonal numbers. To do this, draw pentagons with dots and increase the side length each time. Here are the first few: 2. Write down the first several hexagonal numbers. 3. Make a table for the first ten figurative numbers for the figures: Numbers 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th Triangular Square Pentagonal Hexagonal Heptagonal Octogonal Nonagonal Decagonal What patterns do you see? 2 Figurative Numbers Part 3 1. Can you find a formula for P (n) (the n-th pentagonal number)? 2. Can you find a formula for H(n) (hexagonal numbers)? 3. Explore centered figurative numbers. Here is a picture of what this means: 4. Given a triangular number, find a formula to find which triangular number it is. (This could be called the triangular root of a number, like a square root.) 5. (Fermat Polygonal Number Theorem) (a) For each number 1 through 50, write it as a sum of triangular numbers. Try to use as few numbers as possible. (b) Try to write numbers are sums of square numbers. How many squares do you need? (c) Sums of pentagonal numbers? How many pentagonal numbers do you need? 3.

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Integer Partitions
12 PETER KOROTEEV Remark. Note that out n! fixed points of T acting on Xn only for single point q we 12 PETER KOROTEEV get a polynomial out of Vq.Othercoefficient functions Vp remain infinite series which we 12shall further ignore. PETER One KOROTEEV can verify by examining (2.25) that in the n limit these Remark. Note that out n! fixed points of T acting on Xn only for single point q we coefficient functions will be suppressed. A choice of fixed point q following!1 the large-n limits get a polynomial out of Vq.Othercoefficient functions Vp remain infinite series which we Remark. Note thatcorresponds out n! fixed to zooming points of intoT aacting certain on asymptoticXn only region for single in which pointa q we a shall further ignore. One can verify by examining (2.25) that in the n | Slimit(1)| these| S(2)| ··· get a polynomial out ofaVSq(n.Othercoe) ,whereS fficientSn is functionsa permutationVp remain corresponding infinite to series the choice!1 which of we fixed point q. shall furthercoe ignore.fficient functions One| can| will verify be suppressed. by2 examining A choice (2.25 of) that fixed in point theqnfollowinglimit the large- thesen limits corresponds to zooming into a certain asymptotic region in which a!1 a coefficient functions will be suppressed. A choice of fixed point q following| theS(1) large-| | nSlimits(2)| ··· aS(n) ,whereS Sn is a permutation corresponding to the choice of fixed point q. q<latexit sha1_base64="gqhkYh7MBm9GaiVNDBxVo5M1bBY=">AAAB8HicbVBNS8NAEN3Ur1q/qh69LBbBU0lEUG9FLx4rWFtsQ9lsJ+3SzSbuTsQS+i+8eFDx6s/x5r9x2+agrQ8GHu/NMDMvSKQw6LrfTmFpeWV1rbhe2tjc2t4p7+7dmTjVHBo8lrFuBcyAFAoaKFBCK9HAokBCMxheTfzmI2gjYnWLowT8iPWVCAVnaKX7DsITBmH2MO6WK27VnYIuEi8nFZKj3i1/dXoxTyNQyCUzpu25CfoZ0yi4hHGpkxpIGB+yPrQtVSwC42fTi8f0yCo9GsbalkI6VX9PZCwyZhQFtjNiODDz3kT8z2unGJ77mVBJiqD4bFGYSooxnbxPe0IDRzmyhHEt7K2UD5hmHG1IJRuCN//yImmcVC+q3s1ppXaZp1EkB+SQHBOPnJEauSZ10iCcKPJMXsmbY5wX5935mLUWnHxmn/yB8/kDhq2RBA==</latexit>
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Induction for Tetrahedral Numbers
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Tetrahedral Numbers
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Notations Used 1
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