This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A035681 #17 Jan 03 2025 09:27:46
%S A035681 0,0,0,0,1,1,1,1,2,2,2,2,5,5,5,5,8,8,8,8,14,15,15,15,22,23,23,23,34,
%T A035681 37,38,38,51,54,55,55,74,81,84,85,108,116,119,120,151,165,172,175,214,
%U A035681 230,238,241,291,317,332,339,402,434,451,459,538,584,613,628,727,784
%N A035681 Number of partitions of n into parts 8k+1 and 8k+4 with at least one part of each type.
%H A035681 Robert Price, <a href="/A035681/b035681.txt">Table of n, a(n) for n = 1..1000</a>
%F A035681 G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(8*k + 4))). - _Robert Price_, Aug 15 2020
%t A035681 nmax = 66; s1 = Range[0, nmax/8]*8 + 1; s2 = Range[0, nmax/8]*8 + 4;
%t A035681 Table[Count[IntegerPartitions[n, All, s1~Join~s2], x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 15 2020 *)
%t A035681 nmax = 66; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 15 2020 *)
%Y A035681 Cf. A035441-A035468, A035618-A035680, A035682-A035699.
%K A035681 nonn
%O A035681 1,9
%A A035681 _Olivier Gérard_