A262956 Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n - x^4 - y*(y+1)/2 is a square or a square minus 1.

1, 2, 2, 3, 3, 3, 4, 2, 2, 5, 5, 3, 2, 3, 4, 4, 4, 5, 7, 5, 3, 6, 5, 3, 7, 8, 5, 4, 5, 7, 8, 6, 2, 4, 5, 5, 10, 7, 5, 7, 6, 4, 3, 5, 8, 10, 6, 2, 3, 5, 6, 10, 9, 5, 7, 6, 4, 4, 5, 6, 8, 5, 3, 8, 7, 5, 7, 5, 6, 11, 9, 5, 3, 5, 5, 4, 4, 3, 8, 9, 7, 10, 7, 5, 11, 10, 8, 5, 1

Offset: 1

Author

Zhi-Wei Sun, Oct 05 2015

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0. In other words, for any positive integer n, either n or n + 1 can be written as the sum of a fourth power, a square and a positive triangular number.

We also guess that a(n) = 1 only for n = 1, 89, 244, 464, 5243, 14343.

Examples

			a(1) = 1 since 1 = 0^4 + 1*2/2 + 0^2.
a(89) = 1 since 89 = 2^4 + 4*5/2 + 8^2 - 1.
a(244) = 1 since 244 = 2^4 + 2*3/2 + 15^2.
a(464) = 1 since 464 = 2^4 + 22*23/2 + 14^2 - 1.
a(5243) = 1 since 5243 = 0^4 + 50*51/2 + 63^2 - 1.
a(14343) = 1 since 14343 = 2^4 + 163*164/2 + 31^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Crossrefs

Cf. A000217, A000290, A000583, A262941, A262944, A262945, A262954, A262955, A262959.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]||IntegerQ[Sqrt[n+1]]
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}]