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 A241091 #7 May 11 2025 19:16:28
%S A241091 0,1,1,2,1,2,3,3,3,4,5,5,7,7,9,10,11,12,15,16,19,22,24,27,30,34,37,43,
%T A241091 47,53,59,66,72,82,88,99,109,120,131,146,160,176,194,212,233,256,279,
%U A241091 304,334,362,396,431,471,510,558,604,659,714,776,839,913,985
%N A241091 Number of partitions p of n into distinct parts such that max(p) <= 1 + 2*(number of parts of p).
%F A241091 a(n) = A241086(n) + A241092(n) for n >= 0.
%e A241091 a(12) counts these 5 partitions: 741, 732, 651, 642, 6321, 543, 5421.
%t A241091 z = 30; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t A241091 Table[Count[f[n], p_ /; Max[p] < 1 + 2*Length[p]], {n, 0, z}] (*A241086*)
%t A241091 Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Length[p]], {n, 0, z}](*A241091*)
%t A241091 Table[Count[f[n], p_ /; Max[p] == 1 + 2*Length[p]], {n, 0, z}](*A241092*)
%t A241091 Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Length[p]], {n, 0, z}](*A241089*)
%t A241091 Table[Count[f[n], p_ /; Max[p] > 1 + 2*Length[p]], {n, 0, z}] (*A241093*)
%Y A241091 Cf. A241086, A241092, A241093.
%K A241091 nonn,easy
%O A241091 0,4
%A A241091 _Clark Kimberling_, Apr 18 2014