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%I #27 Mar 13 2015 04:59:20
%S 1,1,1,1,1,1,1,1,2,1,2,2,1,1,1,2,3,2,3,2,1,3,3,3,3,2,4,2,1,3,2,3,4,3,
%T 4,2,2,4,4,3,4,3,6,5,2,4,3,4,5,4,6,4,1,4,5,4,5,5,7,4,1,5,5,5,6,5,8,5,
%U 3,4,6,6,6,6,7,6,3,6,6,5,6,6,10,7,1,5,8,7,7,7,8,5,3,6,7,6,8,7,10,8,3
%N Number of ways to write n as the sum of four unordered generalized octagonal numbers.
%C I have proved that a(n) > 0 for all n, i.e., any nonnegative integer can be expressed as the sum of four generalized octagonal numbers. I can show that a(n) = 1 if 3*n+4 is among 7, 13, 19, 31, 43, 2^(2k), 5*2^(2k+1), 11*2^(2k+1), 23*2^(2k+1) (k = 0,1,2,...), and conjecture the converse.
%C I also conjecture that each nonnegative integer can be written as the sum of two heptagonal numbers, a second heptagonal number and a generalized heptagonal number.
%H Zhi-Wei Sun, <a href="/A255934/b255934.txt">Table of n, a(n) for n = 0..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, arXiv:0905.0635 [math.NT], 2009-2015.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1503.03743">A result similar to Lagrange's theorem</a>, arXiv:1503.03743 [math.NT], 2015.
%e a(60) = 1 since 60 = 1*(3*1-2) + (-1)*(3*(-1)-2) + 3*(3*3-2) + (-3)*(3*(-3)-2).
%e a(1876) = 1 since 1876 = (-5)*(3*(-5)-2) + (-5)*(3*(-5)-2) + 11*(3*11-2) + (-21)*(3*(-21)-2).
%e a(15700) = 1 since 15700 = 11*(3*11-2) + (-21)*(3*(-21)-2) + 43*(3*43-2) + (-53)*(3*(-53)-2).
%e a(21844) = 1 since 21844 = 43*(3*43-2) + 43*(3*43-2) + 43*(3*43-2) + 43*(3*43-2).
%e a(30036) = 1 since 30036 = (-21)*(3*(-21)-2) + (-21)*(3*(-21)-2) + 43*(3*43-2) + (-85)*(3*(-85)-2).
%t T[n_]:=Union[Table[x(3x-2),{x,-Floor[(Sqrt[3n+1]-1)/3],Floor[(Sqrt[3n+1]+1)/3]}]]
%t Do[r=0;Do[If[n-Part[T[n],x]-Part[T[n],y]-Part[T[n],z]<Part[T[n],z],Goto[aa]];If[MemberQ[Table[Part[T[n],w],{w,z,Length[T[n]]}],n-Part[T[n],x]-Part[T[n],y]-Part[T[n],z]]==True,r=r+1];Label[aa];Continue,{x,1,Length[T[n]]},{y,x,Length[T[n]]},{z,y,Length[T[n]]}];Print[n," ",r];Continue,{n,0,100}]
%Y Cf. A000566, A001082, A085787, A147875, A255350.
%K nonn
%O 0,9
%A _Zhi-Wei Sun_, Mar 11 2015
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