A262941 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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Oct 04 2015

Keywords

• nonn

Comments

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.

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.

See also A262944, A262945, A262954, A262955, A262956 for similar conjectures.

Examples

			a(1)    = 1 since    1 = 0^4 +    0^2 +  1*2/2  with  0 even.
a(2)    = 1 since    2 = 1^4 +    0^2 +  1*2/2  with  0 even.
a(13)   = 1 since   13 = 1^4 + 2* 1^2 +  4*5/2.
a(16)   = 1 since   16 = 1^4 +    0^2 +  5*6/2  with  0 even.
a(50)   = 1 since   50 = 1^4 +    2^2 +  9*10/2 with  2 even.
a(59)   = 1 since   59 = 0^4 +    2^2 + 10*11/2 with  2 even.
a(239)  = 1 since  239 = 0^4 + 2* 2^2 + 21*22/2 with  2 even.
a(493)  = 1 since  493 = 2^4 +   18^2 + 17*18/2 with 18 even.
a(1156) = 1 since 1156 = 1^4 + 2*24^2 +  2*3/2  with 24 even.
a(1492) = 1 since 1492 = 2^4 + 2* 7^2 + 52*53/2.
a(1984) = 1 since 1984 = 5^4 +   18^2 + 45*46/2 with 18 even.
a(3332) = 1 since 3332 = 5^4 +   52^2 +  2*3/2  with 52 even.
		

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.
• Zhi-Wei Sun, On universal sums ax^2+by^2+f(z), aT_x+bT_y+f(z) and aT_x+by^2+f(z), arXiv:1502.03056 [math.NT], 2015.

Crossrefs

Cf. A000217, A000290, A000583, A254885, A262813, A262827, A262944, A262945, A262954, A262955, A262956.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n/2]]||IntegerQ[Sqrt[n/4]]
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}]