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 A255350 Number of ways to write n as a*(2a-1)+ b*(2b-1) + c*(2c+1) + d*(2d+1), where a,b,c,d are nonnegative integers with a <= b and c <= d. 9
 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 3, 2, 1, 2, 2, 2, 4, 4, 1, 2, 3, 3, 3, 3, 2, 2, 4, 3, 3, 3, 2, 5, 4, 3, 3, 4, 3, 4, 5, 2, 3, 5, 3, 5, 5, 2, 5, 5, 3, 5, 4, 4, 5, 6, 5, 4, 4, 3, 4, 5, 5, 7, 7, 1, 5, 7, 4, 7, 7, 4, 3, 8, 5, 5, 6, 6, 5, 6, 4, 6, 6, 5, 10, 7, 3, 5, 8, 7, 9, 7, 4, 4, 9, 5, 4, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Conjecture: (i) a(n) > 0 for all n. In other words, any nonnegative integer can be expressed as the sum of two hexagonal numbers and two second hexagonal numbers. (ii) Each nonnegative integer can be written as the sum of two pentagonal numbers and two second pentagonal numbers. We have verified parts (i) and (ii) of the conjecture for n up to 10^7 and 10^6 respectively. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015. Zhi-Wei Sun, On universal sums a*x^2+b*y^2+f(z), a*T_x+b*T_y+f(z) and a*T_x+b*y^2+f(z), arXiv:1502.03056 [math.NT], 2015. EXAMPLE a(23) = 1 since 23 = 1*(2*1-1) + 1*(2*1-1) + 0*(2*0+1) + 3*(2*3+1). a(68) = 1 since 68 = 1*(2*1-1) + 4*(2*4-1) + 1*(2*1+1) + 4*(2*4+1). MATHEMATICA HQ[n_]:=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1], 4]==1 Do[r=0; Do[If[HQ[n-x(2x-1)-y(2y-1)-z(2z+1)], r=r+1], {x, 0, (Sqrt[4n+1]+1)/4}, {y, x, (Sqrt[8(n-x(2x-1))+1]+1)/4}, {z, 0, (Sqrt[4(n-x(2x-1)-y(2y-1))+1]-1)/4}]; Print[n, " ", r]; Continue, {n, 0, 10000}] CROSSREFS Cf. A000326, A000384, A005449, A014105. Sequence in context: A112933 A270650 A088427 * A104482 A333632 A209332 Adjacent sequences:  A255347 A255348 A255349 * A255351 A255352 A255353 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 21 2015 STATUS approved

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Last modified April 15 20:40 EDT 2021. Contains 342977 sequences. (Running on oeis4.)