cp's OEIS Frontend

A254623 Number of ways to write n as x^2 + y*(3*y+1)/2 + z*(5*z+3)/2 with x,y,z nonnegative integers.

%I A254623 #9 Jul 08 2025 07:48:33
%S A254623 1,1,1,1,2,1,2,2,2,1,1,3,1,2,1,4,4,1,1,3,4,1,2,2,3,1,1,4,3,5,3,5,2,1,
%T A254623 2,3,4,1,4,2,5,1,3,5,4,3,3,2,3,4,2,5,2,6,4,5,3,5,2,1,2,3,8,1,6,4,3,2,
%U A254623 3,5,6,5,2,4,2,3,5,6,7,5,1,6,3,4,3,4,8,2,5,5,4,3,3,6,4,4,3,7,1,2,6
%N A254623 Number of ways to write n as x^2 + y*(3*y+1)/2 + z*(5*z+3)/2 with x,y,z nonnegative integers.
%C A254623 Conjecture: (i) a(n) > 0 for all n, and a(n) > 1 for all n > 118.
%C A254623 (ii) For each m = 5,7,8, any nonnegative integer n can be written as the sum of two triangular numbers and a second m-gonal number, where the second m-gonal numbers are given by (m-2)*k*(k+1)/2-k (k = 0,1,...).
%C A254623 (iii) For every m = 5,6,7,9,11, any nonnegative integer n can be written as the sum of a triangular number, a square and a second m-gonal number.
%C A254623 Note that k*(3*k+1)/2 (k = 0,1,...) are second pentagonal numbers and k*(5*k+3)/2 (k = 0,1,...) are second heptagonal numbers. The conjecture has been verified for all n = 0.. 2*10^6.
%H A254623 Zhi-Wei Sun, <a href="/A254623/b254623.txt">Table of n, a(n) for n = 0..10000</a>
%H A254623 Zhi-Wei Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, arXiv:1405.0635 [math.NT], 2009-2015.
%e A254623 a(41) = 1 since 41 = 1^2 + 5*(3*5+1)/2 + 0*(5*0+3)/2.
%e A254623 a(98) = 1 since 98 = 8^2 + 2*(3*2+1)/2 + 3*(5*3+3)/2.
%e A254623 a(118) = 1 since 118 = 2^2 + 3*(3*3+1)/2 + 6*(5*6+3)/2.
%t A254623 SQ[n_]:=IntegerQ[Sqrt[n]]
%t A254623 Do[r=0;Do[If[SQ[n-y(3y+1)/2-z(5z+3)/2],r=r+1],{y,0,(Sqrt[24n+1]-1)/6},{z,0,(Sqrt[40(n-y(3y+1)/2)+9]-3)/10}];
%t A254623 Print[n," ",r];Continue,{n,0,100}]
%Y A254623 Cf. A000217, A000290, A005449, A147875.
%K A254623 nonn
%O A254623 0,5
%A A254623 _Zhi-Wei Sun_, Feb 03 2015