%I
%S 1,2,2,2,2,2,3,2,1,4,5,3,1,2,4,3,3,3,5,4,2,4,5,2,3,6,4,3,4,5,5,4,3,2,
%T 5,4,7,7,3,4,3,4,2,4,6,6,6,2,2,2,4,5,9,5,4,5,2,3,2,5,5,5,2,4,5,3,4,5,
%U 4,5,7,3,3,3,6,3,4,4,5,6,3,7,7,3,4,8,7,7,1,3,9,8,6
%N Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n  x^4  y*(y+1)/2 has the form z^2 or 8*z^2.
%C Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 9, 13, 89, 449, 524, 1049, 2164, 14969, 51334.
%C (ii) For any positive integer n, there are integers x and y > 0 such that n  x^4  T(y) has the form T(z) or 4*T(z), where T(k) refers to the triangular number k*(k+1)/2.
%C (iii) For every n = 1,2,3,... there are integers x and y > 0 such that n  x^4  T(y) has the form T(z) or 2*z^2.
%C (iv) For {c,d} = {1,2} and n > 0, there are integers x and y > 0 such that n  2*x^4  T(y) has the form c*T(z) or d*z^2.
%C (v) For each n = 1,2,3,... there are integers x and y > 0 such that n  4*x^4  T(y) has the form 2*T(z) or z^2.
%C See also A262941, A262944, A262945, A262954, A262955 and A262956 for similar conjectures.
%H ZhiWei Sun, <a href="/A262954/b262954.txt">Table of n, a(n) for n = 1..10000</a>
%H ZhiWei Sun, <a href="http://dx.doi.org/10.4064/aa12721">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127(2007), 103113.
%e a(1) = 1 since 1 = 0^4 + 1*2/2 + 0^2.
%e a(9) = 1 since 9 = 0^4 + 1*2/2 + 8*1^2.
%e a(13) = 1 since 13 = 1^4 + 2*3/2 + 3^2.
%e a(89) = 1 since 89 = 2^4 + 1*2/2 + 8*3^2.
%e a(449) = 1 since 449 = 0^4 + 22*23/2 + 14^2.
%e a(524) = 1 since 524 = 3^4 + 29*30/2 + 8*1^2.
%e a(1049) = 1 since 1049 = 5^4 + 16*17/2 + 8*6^2.
%e a(2164) = 1 since 2164 = 1^4 + 34*35/2 + 8*14^2.
%e a(14969) = 1 since 14969 = 8^4 + 145*146/2 + 8*6^2.
%e a(51334) = 1 since 51334 = 5^4 + 313*314/2 + 8*14^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]IntegerQ[Sqrt[n/8]]
%t Do[r=0;Do[If[SQ[nx^4y(y+1)/2],r=r+1],{x,0,n^(1/4)},{y,1,(Sqrt[8(nx^4)+1]1)/2}];Print[n," ",r];Continue,{n,1,100}]
%Y Cf. A000217, A000290, A000583, A262941, A262944, A262945, A262954, A262955, A262956.
%K nonn
%O 1,2
%A _ZhiWei Sun_, Oct 05 2015
