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

%I A303825 #19 Oct 19 2019 10:15:53
%S A303825 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,
%T A303825 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,
%U A303825 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
%N A303825 Number of ways of writing n as a sum of powers of 7, each power being used at most seven times.
%H A303825 Seiichi Manyama, <a href="/A303825/b303825.txt">Table of n, a(n) for n = 0..10000</a>
%F A303825 G.f.: Product_{k>=0} (1-x^(8*7^k))/(1-x^(7^k)).
%F A303825 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.
%F A303825 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
%p A303825 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
%p A303825       add(b(n-j*7^i, i-1), j=0..min(7, n/7^i))))
%p A303825     end:
%p A303825 a:= n-> b(n, ilog[7](n)):
%p A303825 seq(a(n), n=0..120);  # _Alois P. Heinz_, May 01 2018
%t A303825 m = 120; A[_] = 1;
%t A303825 Do[A[x_] = Total[x^Range[0, 7]] A[x^7] + O[x]^m // Normal, {m}];
%t A303825 CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 19 2019 *)
%o A303825 (Ruby)
%o A303825 def A(k, n)
%o A303825   ary = [1]
%o A303825   (1..n).each{|i|
%o A303825     s = ary[i / k]
%o A303825     s += ary[i / k - 1] if i % k == 0
%o A303825     ary << s
%o A303825   }
%o A303825   ary
%o A303825 end
%o A303825 p A(7, 100)
%Y A303825 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).
%K A303825 nonn
%O A303825 0,8
%A A303825 _Seiichi Manyama_, May 01 2018