A255191 - OEIS

%I #11 Feb 16 2015 11:20:14

%S 1,1,1,1,2,1,3,2,1,1,2,2,2,2,1,2,4,2,2,2,4,2,4,1,2,4,2,2,2,3,3,4,3,1,

%T 2,2,4,4,4,2,4,2,5,4,1,3,6,4,2,3,3,3,5,2,2,5,5,3,3,3,3,4,3,1,5,5,4,6,

%U 2,2,6,4,6,5,4,2,6,3,3,3,5,5,6,3,2,7,2,4,4,2,5,6,6,3,4,5,2,6,3,2,6

%N Number of solutions to the equation n = x^2 + y*(y+1)/2 + z*(z+1)/2 (x,y,z=0,1,...) with x == Floor[sqrt(n)] (mod 2) and y <= z.

%C By Dickson's book in the reference, E. Lionnet observed in 1872 that any nonnegative integer can be written as the sum of a square and two triangular numbers.

%C We have a(n) > 0 for all n. This follows from Theorem 1(i) and Lemma 2 of Sun's reference in 2007. Note that if n = m*(m+1) with m a nonnegative integer then floor(sqrt(n)) = m.

%D L. E. Dickson, History of the Theory of Numbers, vol. II, AMS Chelsea Publ., 1999, p. 24.

%H Zhi-Wei Sun, <a href="/A255191/b255191.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1502.03056">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)</a>, arXiv:1502.03056 [math.NT], 2015.

%e a(14) = 1 since 14 = 1^2 + 2*3/2 + 4*5/2 with floor(sqrt(14)) == 1 (mod 2).

%e a(44) = 1 since 44 = 4^2 + 0*1/2 + 7*8/2 with floor(sqrt(44)) == 4 (mod 2).

%e a(63) = 1 since 63 = 5^2 + 4*5/2 + 7*8/2 with floor(sqrt(63)) == 5 (mod 2).

%t TQ[n_]:=IntegerQ[Sqrt[8n+1]]

%t Do[r=0;Do[If[Mod[Floor[Sqrt[n]]-x,2]==0&&TQ[n-x^2-y*(y+1)/2], r=r+1],{x,0,Sqrt[n]},{y,0,(Sqrt[4(n-x^2)+1]-1)/2}];

%t Print[n," ",r];Label[aa];Continue,{n,0,100}]

%Y Cf. A000217, A000290, A254885.

%K nonn

%O 0,5

%A _Zhi-Wei Sun_, Feb 16 2015