A303825 Number of ways of writing n as a sum of powers of 7, each power being used at most seven times.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 2, 2, 2, 2, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 2, 2, 2, 2, 2, 3

Offset

0,8

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{k>=0} (1-x^(8*7^k))/(1-x^(7^k)).

a(0)=1; for k>0, a(7*k) = a(k)+a(k-1) and a(7*k+r) = a(k) with r=1,2,3,4,5,6.

G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7) * A(x^7). - Ilya Gutkovskiy, Jul 09 2019

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add(b(n-j*7^i, i-1), j=0..min(7, n/7^i))))
    end:
a:= n-> b(n, ilog[7](n)):
seq(a(n), n=0..120);  # Alois P. Heinz, May 01 2018

Mathematica

m = 120; A[_] = 1;
Do[A[x_] = Total[x^Range[0, 7]] A[x^7] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 19 2019 *)

Prog

(Ruby)
def A(k, n)
  ary = [1]
  (1..n).each{|i|
    s = ary[i / k]
    s += ary[i / k - 1] if i % k == 0
    ary << s
  }
  ary
end
p A(7, 100)

Crossrefs

Number of ways of writing n as a sum of powers of b, each power being used at most b times: A054390 (b=3), A277872 (b=4), A277873 (b=5), A303824 (b=6), this sequence (b=7).

Sequence in context: A139551 A022931 A373217 * A043280 A030379 A030392

Adjacent sequences: A303822 A303823 A303824 * A303826 A303827 A303828

Keyword

nonn

Author

Seiichi Manyama, May 01 2018

Status

approved