A308566 Number of ways to write n as w^2 + x*(x+1) + 4^y*5^z with w,x,y,z nonnegative integers.

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

Offset: 1

Author

Zhi-Wei Sun, Jun 07 2019

Keywords

• nonn

Comments

Recall an observation of Euler: {w^2 + x*(x+1): w,x = 0,1,2,...} = {a*(a+1)/2 + b*(b+1)/2: a,b = 0,1,...}.

Conjecture: a(n) > 0 for all n > 0. Equivalently, each n = 1,2,3,... can be written as a*(a+1)/2 + b*(b+1)/2 + 4^c*5^d with a,b,c,d nonnegative integers.

See also A308584 for a similar conjecture.

We have verified a(n) > 0 for all n = 1..5*10^8.

a(n) > 0 for 0 < n < 10^10. - Giovanni Resta, Jun 08 2019

Examples

			a(1) = 1 with 1 = 0^2 + 0*1 + 4^0*5^0.
a(2) = 1 with 2 = 1^2 + 0*1 + 4^0*5^0.
a(3) = 1 with 3 = 0^2 + 1*2 + 4^0*5^0.
a(9) = 1 with 9 = 2^2 + 0*1 + 4^0*5^1.
a(303) = 1 with 303 = 16^2 + 6*7 + 4^0*5^1.
a(585) = 1 with 585 = 5^2 + 15*16 + 4^3*5^1.
a(37863) = 2 with 37863 = 166^2 + 101*102 + 4^0*5^1 = 179^2 + 26*27 + 4^5*5^1.
		

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, A000302, A000351, A002378, A303656, A303637, A308411, A308547, A308584.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-4^k*5^m-x(x+1)],r=r+1],{k,0,Log[4,n]},{m,0,Log[5,n/4^k]},{x,0,(Sqrt[4(n-4^k*5^m)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]