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%I #20 Feb 18 2024 01:18:23
%S 1,1,2,2,2,3,2,2,2,3,3,2,1,2,3,1,3,3,6,3,4,4,4,4,3,4,2,3,3,4,3,2,5,3,
%T 4,3,6,5,6,4,2,3,2,4,4,4,5,3,3,1,3,5,6,6,4,3,3,4,1,5,4,3,4,3,4,3,4,4,
%U 5,3,5,4,5,4,4,3,2,4,6,3,4,6,4,5,2,7,7,4,3,3,5,4,5,6,6,5,2,6,4,8
%N Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is an even square or twice a square.
%C Conjecture: a(n) > 0 for all n > 0. In other words, any positive integer n can be written as x^4 + 2^k*y^2 + z*(z+1)/2, where k is 1 or 2, and x,y,z are integers with z > 0.
%C This has been verified for n up to 2*10^6. We also guess that a(n) = 1 only for n = 1, 2, 13, 16, 50, 59, 239, 493, 1156, 1492, 1984, 3332.
%C See also A262944, A262945, A262954, A262955, A262956 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A262941/b262941.txt">Table of n, a(n) for n = 1..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 ax^2+by^2+f(z), aT_x+bT_y+f(z) and aT_x+by^2+f(z)</a>, arXiv:1502.03056 [math.NT], 2015.
%e a(1) = 1 since 1 = 0^4 + 0^2 + 1*2/2 with 0 even.
%e a(2) = 1 since 2 = 1^4 + 0^2 + 1*2/2 with 0 even.
%e a(13) = 1 since 13 = 1^4 + 2* 1^2 + 4*5/2.
%e a(16) = 1 since 16 = 1^4 + 0^2 + 5*6/2 with 0 even.
%e a(50) = 1 since 50 = 1^4 + 2^2 + 9*10/2 with 2 even.
%e a(59) = 1 since 59 = 0^4 + 2^2 + 10*11/2 with 2 even.
%e a(239) = 1 since 239 = 0^4 + 2* 2^2 + 21*22/2 with 2 even.
%e a(493) = 1 since 493 = 2^4 + 18^2 + 17*18/2 with 18 even.
%e a(1156) = 1 since 1156 = 1^4 + 2*24^2 + 2*3/2 with 24 even.
%e a(1492) = 1 since 1492 = 2^4 + 2* 7^2 + 52*53/2.
%e a(1984) = 1 since 1984 = 5^4 + 18^2 + 45*46/2 with 18 even.
%e a(3332) = 1 since 3332 = 5^4 + 52^2 + 2*3/2 with 52 even.
%t SQ[n_]:=IntegerQ[Sqrt[n/2]]||IntegerQ[Sqrt[n/4]]
%t Do[r=0;Do[If[SQ[n-x^4-y(y+1)/2],r=r+1],{x,0,n^(1/4)},{y,1,(Sqrt[8(n-x^4)+1]-1)/2}];Print[n," ",r];Continue,{n,1,100}]
%Y Cf. A000217, A000290, A000583, A254885, A262813, A262827, A262944, A262945, A262954, A262955, A262956.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Oct 04 2015
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