cp's OEIS Frontend

A297199 a(n) = number of partitions of n into consecutive positive cubes.

%I A297199 #32 Aug 22 2019 20:34:33
%S A297199 1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
%T A297199 1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,
%U A297199 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A297199 a(n) = number of partitions of n into consecutive positive cubes.
%H A297199 Antti Karttunen, <a href="/A297199/b297199.txt">Table of n, a(n) for n = 1..65537</a> (terms 1..10000 from Robert Israel)
%F A297199 a(A217843(n)) >= 1 for n > 1.
%F A297199 a(n) >= 2 for n in A265845. - _Robert Israel_, Jan 15 2018
%F A297199 G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k^3). - _Ilya Gutkovskiy_, Apr 18 2019
%F A297199 a(A000578(n)) = A307609(n). - _Antti Karttunen_, Aug 22 2019
%e A297199     1 = 1^3,                   so   a(1) = 1.
%e A297199     8 = 2^3,                   so   a(8) = 1.
%e A297199     9 = 1^3 + 2^3,             so   a(9) = 1.
%e A297199    27 = 3^3,                   so  a(27) = 1.
%e A297199    35 = 2^3 + 3^3,             so  a(35) = 1.
%e A297199    36 = 1^3 + 2^3 + 3^3,       so  a(36) = 1.
%e A297199    64 = 4^3,                   so  a(64) = 1.
%e A297199    91 = 3^3 + 4^3,             so  a(91) = 1.
%e A297199    99 = 2^3 + 3^3 + 4^3,       so  a(99) = 1.
%e A297199   100 = 1^3 + 2^3 + 3^3 + 4^3, so a(100) = 1.
%p A297199 N:= 200: # to get a(1)..a(N)
%p A297199 F:= (a, b) -> (b^2*(b+1)^2-a^2*(a-1)^2)/4:
%p A297199 A:= Vector(N):
%p A297199 for b from 1 to floor(N^(1/3)) do
%p A297199   for a from b to 1 by -1 do
%p A297199      v:= F(a,b);
%p A297199      if v > N then break fi;
%p A297199      A[v]:= A[v]+1;
%p A297199 od od:
%p A297199 convert(A,list); # _Robert Israel_, Jan 15 2018, corrected Jan 29 2018
%o A297199 (PARI) A297199(n) = { my(s=0, k=1, c); while((c=k^3) <= n, my(u=n-c, i=k); while(u>0, i++; c = i^3; u=u-c); s += (!u); k++); (s); }; \\ _Antti Karttunen_, Aug 22 2019
%Y A297199 Cf. A000578, A001227, A062682, A217843, A265845, A296338, A307609.
%K A297199 nonn
%O A297199 1,216
%A A297199 _Seiichi Manyama_, Jan 15 2018