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A160325
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Number of ways to express n=0,1,2,... as the sum of a triangular number, an even square and a pentagonal number.
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3
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1, 2, 1, 1, 2, 3, 3, 2, 2, 1, 3, 3, 2, 1, 1, 5, 3, 3, 2, 4, 3, 2, 6, 2, 2, 2, 5, 4, 3, 3, 1, 4, 4, 3, 1, 1, 5, 7, 5, 3, 4, 6, 4, 3, 4, 5, 2, 3, 3, 5, 4, 5, 5, 2, 6, 2, 5, 5, 5, 3, 3, 6, 3, 2, 5, 4, 6, 6, 3, 3, 6, 9, 6, 5, 4, 5, 5, 6, 2, 7, 4, 3, 6, 6, 4, 2, 7, 7, 3, 3, 4, 5, 8, 5, 5, 5, 8, 4, 2, 4, 6, 6, 7, 6, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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. It is known that any positive integer can be written as the sum of a triangular number, a square and an even square (or an odd square).
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REFERENCES
| 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
| Zhi-Wei Sun, Table of n, a(n) for n = 0..60000
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
| a(n)=|{<x,y,z>: x,y,z=0,1,2,... & x(x+1)/2+4y^2+(3z^2-z)/2}|
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EXAMPLE
| For n=15 the a(15)=5 solutions are 3+0+12, 6+4+5, 10+0+5, 10+4+1, 15+0+0.
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MATHEMATICA
| SQ[x_]:=x>-1&&IntegerPart[Sqrt[x]]^2==x RN[n_]:=Sum[If[SQ[8(n-4y^2-(3z^2-z)/2)+1], 1, 0], {y, 0, Sqrt[n/4]}, {z, 0, Sqrt[n-4y^2]}] Do[Print[n, " ", RN[n]], {n, 0, 60000}]
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CROSSREFS
| A000217, A000290, A000326, A160324
Sequence in context: A128495 A113136 A156267 * A054989 A051631 A073725
Adjacent sequences: A160322 A160323 A160324 * A160326 A160327 A160328
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KEYWORD
| nonn
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AUTHOR
| Zhi-Wei Sun (zwsun(AT)nju.edu.cn), May 08 2009
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EXTENSIONS
| More terms copied from author's b-file by Hagen von Eitzen (math(AT)von-eitzen.de), Jul 20 2009
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