A303827 Number of ways of writing n as a sum of powers of 4, each power being used at most 5 times.

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 2, 2, 4, 4, 2, 2, 3, 3, 1, 1, 2, 2, 1, 1, 3, 3, 2, 2, 4, 4, 2, 2, 3, 3, 1, 1, 2, 2, 1, 1, 3, 3, 2, 2, 4, 4, 2, 2, 3, 3, 1, 1, 2, 2, 1, 1, 4, 4, 3, 3, 6, 6, 3, 3, 5, 5, 2, 2, 4, 4, 2, 2, 6, 6, 4, 4, 8, 8, 4, 4, 6, 6

Offset: 0

Author

Seiichi Manyama, May 01 2018

Keywords

• nonn
• look

Examples

			a(17) = 3 because 17=16+1=4+4+4+4+1=4+4+4+1+1+1+1+1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..16383

Crossrefs

Number of ways of writing n as a sum of powers of b, each power being used at most b+1 times: A117535 (b=3), this sequence (b=4), A303828 (b=5).

Cf. A277872.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add(b(n-j*4^i, i-1), j=0..min(5, n/4^i))))
    end:
a:= n-> b(n, ilog[4](n)):
seq(a(n), n=0..120);  # Alois P. Heinz, May 01 2018

Mathematica

m = 100; A[_] = 1;
Do[A[x_] = (1+x+x^2+x^3+x^4+x^5) * A[x^4] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 06 2019, after Ilya Gutkovskiy *)

Formula

G.f.: Product_{k>=0} (1-x^(6*4^k))/(1-x^(4^k)).

G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3 + x^4 + x^5) * A(x^4). - Ilya Gutkovskiy, Jul 09 2019