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Search: id:A160326
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| A160326 |
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Number of ways to express n=0,1,2,... as the sum of two squares and a pentagonal number. |
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+0
2
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| 1, 3, 3, 1, 2, 5, 4, 1, 1, 5, 6, 2, 1, 5, 5, 2, 4, 6, 5, 1, 3, 6, 5, 3, 1, 8, 8, 4, 2, 4, 8, 4, 5, 1, 4, 5, 4, 10, 6, 6, 5, 8, 6, 1, 3, 6, 6, 4, 6, 4, 7, 8, 8, 8, 5, 7, 4, 4, 6, 5, 6, 8, 7, 4, 8, 8, 6, 5, 4, 7, 7, 8, 7, 7, 8, 8, 8, 7, 3, 4, 12, 4, 4, 7, 3, 13, 12, 12, 5, 2, 12, 4, 5, 6, 6, 8, 10, 8, 3, 5, 11
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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In April 2009, Zhi-Wei Sun conjecturted that a(n)>0 for every n=0,1,2,3,... Note that pentagonal numbers are more sparse than squares. The Gauss-Legendre theorem asserts that n is the sum of three squares if and only if it is not of the form 4^a(8b+7) (a,b=0,1,2,...).
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REFERENCES
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B. K. Oh and Z. W. Sun, Mixed sums of squares and triangular numbers (III), J. Number Theory 129(2009), 964-969.
Z. W. Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
Z. W. Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635. http://arxiv.org/abs/0905.0635
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LINKS
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Zhi-Wei Sun, Table of n, a(n) for n = 0..50000
Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), 2009.
Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).
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FORMULA
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a(n)=|{<x,y,z>: x,y=0,1,2,... & x^2+y^2+(3z^2-z)/2=n}|
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EXAMPLE
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For n=5 the a(5)=5 solutions are 0+0+5, 1+4+5, 4+1+5, 0+4+1, 4+0+1.
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MATHEMATICA
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SQ[x_]:=x>-1&&IntegerPart[Sqrt[x]]^2==x RN[n_]:=Sum[If[SQ[n-y^2-(3z^2-z)/2], 1, 0], {y, 0, Sqrt[n]}, {z, 0, Sqrt[n-y^2]}] Do[Print[n, " ", RN[n]], {n, 0, 50000}]
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CROSSREFS
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A000290, A000326, A160324, A160325
Sequence in context: A096995 A010264 A089680 this_sequence A123561 A133695 A138114
Adjacent sequences: A160323 A160324 A160325 this_sequence A160327 A160328 A160329
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KEYWORD
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nonn
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AUTHOR
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Zhi-Wei Sun (zwsun(AT)nju.edu.cn), May 08 2009
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EXTENSIONS
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More terms copied from the b-file by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 01 2009
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