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 A160326 Number of ways to express n=0,1,2,... as the sum of two squares and a pentagonal number. 10
 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; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS In April 2009, Zhi-Wei Sun conjectured 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,...). LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..50000 B. K. Oh and Z. W. Sun, Mixed sums of squares and triangular numbers (III), J. Number Theory 129(2009), 964-969. Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113. 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). Z. W. Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015. FORMULA a(n) = |{: x,y=0,1,2,... & x^2+y^2+(3z^2-z)/2=n}|. EXAMPLE For n=5 the a(5)=5 solutions are 0+0+5, 1+4+5, 4+1+5, 0+4+1, 4+0+1. MATHEMATICA 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}] CROSSREFS Cf. A000290, A000326, A160324, A160325. Sequence in context: A089680 A309507 A306690 * A213662 A213657 A215596 Adjacent sequences:  A160323 A160324 A160325 * A160327 A160328 A160329 KEYWORD nonn AUTHOR Zhi-Wei Sun, May 08 2009 STATUS approved

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Last modified April 1 05:04 EDT 2020. Contains 333155 sequences. (Running on oeis4.)