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 A275824 Number of ways to write n as pen(x) + pen(y) + pen (z) + 2*pen(w), where x,y,z,w are nonnegative integers with x <= y <= z, and pen(k) denotes the pentagonal number k*(3*k-1)/2. 1
 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 3, 3, 2, 3, 2, 3, 1, 1, 2, 1, 3, 2, 6, 4, 3, 4, 2, 3, 2, 2, 3, 1, 5, 4, 4, 5, 4, 4, 1, 3, 3, 1, 3, 5, 7, 6, 4, 5, 3, 5, 3, 3, 4, 4, 5, 4, 6, 7, 3, 5, 4, 4, 3, 2, 6, 3, 6, 5, 5, 7, 7, 7, 3, 6, 7, 5, 4, 4, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Conjecture: (i) a(n) > 0 for all n = 0,1,2,...., and a(n) = 1 only for n = 0, 1, 4, 6, 8, 9, 18, 19, 21, 33, 40, 43. (ii) For every m = 5,6,7,... and each integer d > 1 not divisible by 4, any sufficiently large integer can be written as x + y + z + d*w (or x + y + 2*z + d*w) with x,y,z,w m-gonal numbers. Part (i) of the conjecture is stronger than the classical result that any natural number is the sum of five pentagonal numbers. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 Xiang-Zi Meng and Zhi-Wei Sun, Sums of four polygonal numbers with coefficients, arXiv:1608.02022 [math.NT], 2016. Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211. EXAMPLE a(4) = 1 since 4 = 0*(3*0-1)/2 + 1*(3*1-1)/2 + 1*(3*1-1)/2 + 1*(3*1-1). a(18) = 1 since 18 = 1*(3*1-1)/2 + 2*(3*2-1)/2 + 3*(3*3-1)/2 + 0*(3*0-1). a(19) = 1 since 19 = 0*(3*0-1)/2 + 2*(3*2-1)/2 + 3*(3*3-1)/2 + 1*(3*1-1). a(21) = 1 since 21 = 1*(3*1-1)/2 + 2*(3*2-1)/2 + 2*(3*2-1)/2 + 2*(3*2-1). a(33) = 1 since 33 = 0*(3*0-1)/2 + 1*(3*1-1)/2 + 4*(3*4-1)/2 + 2*(3*2-1). a(40) = 1 since 40 = 0*(3*0-1)/2 + 2*(3*2-1)/2 + 5*(3*5-1)/2 + 0*(3*0-1). a(43) = 1 since n = 1*(3*1-1)/2 + 2*(3*2-1)/2 + 5*(3*5-1)/2 + 1*(3*1-1). MATHEMATICA SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]] pQ[n_]:=pQ[n]=SQ[24n+1]&&(n==0||Mod[Sqrt[24n+1]+1, 6]==0) Do[r=0; Do[If[pQ[n-x*(3x-1)-y(3y-1)/2-z(3z-1)/2], r=r+1], {x, 0, (Sqrt[12n+1]+1)/6}, {y, 0, (Sqrt[8(n-x*(3x-1))+1]+1)/6}, {z, y, (Sqrt[12(n-x*(3x-1)-y(3y-1)/2)+1]+1)/6}]; Print[n, " ", r]; Continue, {n, 0, 80}] CROSSREFS Cf. A000326, A256106. Sequence in context: A289014 A193238 A323826 * A324869 A167678 A078614 Adjacent sequences:  A275821 A275822 A275823 * A275825 A275826 A275827 KEYWORD nonn AUTHOR Zhi-Wei Sun, Aug 10 2016 STATUS approved

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Last modified September 22 00:26 EDT 2020. Contains 337275 sequences. (Running on oeis4.)