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 A160325 Number of ways to express n=0,1,2,... as the sum of a triangular number, an even square and a pentagonal number. 16
 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; 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. 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). LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..60000 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,z=0,1,2,... & x(x+1)/2+4y^2+(3z^2-z)/2}|. 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. 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}] CROSSREFS Cf. A000217, A000290, A000326, A160324. Sequence in context: A326281 A113136 A156267 * A054989 A051631 A073725 Adjacent sequences:  A160322 A160323 A160324 * A160326 A160327 A160328 KEYWORD nonn AUTHOR Zhi-Wei Sun, May 08 2009 EXTENSIONS More terms copied from author's b-file by Hagen von Eitzen, Jul 20 2009 STATUS approved

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Last modified April 9 14:04 EDT 2020. Contains 333353 sequences. (Running on oeis4.)