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A303824 Number of ways of writing n as a sum of powers of 6, each power being used at most six times.

%I A303824 #19 Oct 19 2019 10:15:32
%S A303824 1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,
%T A303824 1,1,3,2,2,2,2,2,3,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,2,1,
%U A303824 1,1,1,1,3,2,2,2,2,2,3,1,1,1,1,1,2,1,1,1,1,1,2
%N A303824 Number of ways of writing n as a sum of powers of 6, each power being used at most six times.
%H A303824 Seiichi Manyama, <a href="/A303824/b303824.txt">Table of n, a(n) for n = 0..10000</a>
%F A303824 G.f.: Product_{k>=0} (1-x^(7*6^k))/(1-x^(6^k)).
%F A303824 a(0)=1; for k>0, a(6*k) = a(k)+a(k-1) and a(6*k+r) = a(k) with r=1,2,3,4,5.
%F A303824 G.f. A(x) satisfies: A(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6) * A(x^6). - _Ilya Gutkovskiy_, Jul 09 2019
%p A303824 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
%p A303824       add(b(n-j*6^i, i-1), j=0..min(6, n/6^i))))
%p A303824     end:
%p A303824 a:= n-> b(n, ilog[6](n)):
%p A303824 seq(a(n), n=0..120);  # _Alois P. Heinz_, May 01 2018
%t A303824 m = 100; A[_] = 1;
%t A303824 Do[A[x_] = Total[x^Range[0, 6]] A[x^6] + O[x]^m // Normal, {m}];
%t A303824 CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 19 2019 *)
%o A303824 (Ruby)
%o A303824 def A(k, n)
%o A303824   ary = [1]
%o A303824   (1..n).each{|i|
%o A303824     s = ary[i / k]
%o A303824     s += ary[i / k - 1] if i % k == 0
%o A303824     ary << s
%o A303824   }
%o A303824   ary
%o A303824 end
%o A303824 p A(6, 100)
%Y A303824 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), this sequence (b=6), A303825 (b=7).
%K A303824 nonn
%O A303824 0,7
%A A303824 _Seiichi Manyama_, May 01 2018