cp's OEIS Frontend

A254885 Number of ways to write n as the sum of two squares and a positive triangular number.

%I A254885 #22 Feb 15 2015 00:02:26
%S A254885 1,1,2,1,2,2,2,2,1,3,4,2,1,3,3,3,2,2,5,3,3,2,5,2,2,5,2,5,3,4,4,4,3,1,
%T A254885 6,3,5,5,3,5,5,3,2,5,3,8,5,2,3,4,5,3,8,4,7,6,3,3,4,5,5,6,3,5,7,4,4,8,
%U A254885 2,6,9,2,6,6,6,4,4,5,6,7,5,6,6,4,4,11,4,6,5,3,9,6,5,4,11,6,3,4,3,9
%N A254885 Number of ways to write n as the sum of two squares and a positive triangular number.
%C A254885 We have shown that a(n) > 0 for all n > 0. In fact, if n is a positive triangular number T(x) = x*(x+1)/2, then n = 0^2 + 0^2 + T(x); if n > 0 is not a triangular number, then by Theorem 1(ii) of the reference of Sun in 2007, there are nonnegative integers a,b,c,u,v,w such that n = a^2 + b^2 + T(c) = u^2 + v^2 + T(w) with a + b odd and u + v even, hence c and w cannot both be zero.
%C A254885 This result is stronger than Euler's observation that any nonnegative integer can be written as the sum of two squares and a triangular number. We have also proved that any positive integer can be written as the sum of a positive square and two triangular numbers.
%H A254885 Zhi-Wei Sun, <a href="/A254885/b254885.txt">Table of n, a(n) for n = 1..10000</a>
%H A254885 Zhi-Wei Sun, <a href="http://dx.doi.org/10.4064/aa127-2-1">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127(2007), 103-113.
%H A254885 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1502.03056">On universal sums a*x^2+b*y^2+c*f(z), a*T_x+b*T_y+f(z) and a*T_x+b*y^2+c*f(z)</a>, arXiv:1502.03056 [math.NT], 2015.
%e A254885 a(4) = 1 since 4 = 0^2 + 1^2 + 2*3/2.
%e A254885 a(9) = 1 since 9 = 2^2 + 2^2 + 1*2/2.
%e A254885 a(13) = 1 since 13 = 1^2 + 3^2 + 2*3/2.
%e A254885 a(34) = 1 since 34 = 2^2 + 3^2 + 6*7/2.
%t A254885 TQ[n_]:=n>0&&IntegerQ[Sqrt[8n+1]]
%t A254885 Do[r=0;Do[If[TQ[n-x^2-y^2],r=r+1],{x,0,Sqrt[n]},{y,0,x}];
%t A254885 Print[n," ",r];Continue,{n,1,10000}]
%Y A254885 Cf. A000217, A000290.
%K A254885 nonn
%O A254885 1,3
%A A254885 _Zhi-Wei Sun_, Feb 10 2015