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
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
Programs
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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