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 A114091 #13 Sep 03 2022 08:29:28
%S A114091 1,1,2,2,2,4,3,3,7,4,4,11,5,5,16,6,6,22,7,7,29,8,8,37,9,9,46,10,10,56,
%T A114091 11,11,67,12,12,79,13,13,92,14,14,106,15,15,121,16,16,137,17,17,154,
%U A114091 18,18,172,19,19,191,20,20,211,21,21,232,22,22,254,23,23,277,24
%N A114091 Number of partitions of n into parts that are distinct mod 3.
%C A114091 Each partition can have at most three parts if n is a multiple of three and at most two parts otherwise. - _Andrew Howroyd_, Jan 28 2020
%C A114091 In general, these sequences can be generated by a linear recurrence with a signature that contains k=1..d tuples of the form (d-1 zeros, (-1)^(k-1) * binomial(d, k)), where d = number of distinct parts (here: d=3). - _Georg Fischer_, Sep 03 2022
%H A114091 Andrew Howroyd, <a href="/A114091/b114091.txt">Table of n, a(n) for n = 1..1000</a>
%H A114091 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).
%F A114091 a(3*n) = 1 + n + binomial(n, 2); a(3*n-1) = a(3*n-2) = n. - _Andrew Howroyd_, Jan 28 2020
%e A114091 a(5)=2 because there are 2 such partition of 5: {5}, {2,3}.
%t A114091 << DiscreteMath`Combinatorica`; np[n_]:= Length@Select[Mod[ #,3]& /@ Partitions[n],(Length@# != Length@Union@#)&]; lst = Array[np,50] (* or *)
%t A114091 LinearRecurrence[{0, 0, 3, 0, 0, -3, 0, 0, 1}, {1, 1, 2, 2, 2, 4, 3, 3, 7}, 64] (* _Georg Fischer_, Sep 03 2022 *)
%o A114091 (PARI) a(n)={1 + n\3 + if(n%3==0, binomial(n/3,2))} \\ _Andrew Howroyd_, Jan 28 2020
%Y A114091 Cf. A008619(d=2), A114092(4), A114093(5), A114094(6), A114095(7), A114096(8), A114098(9), A114097(10).
%K A114091 nonn
%O A114091 1,3
%A A114091 _Giovanni Resta_, Feb 06 2006
%E A114091 Terms a(51) and beyond from _Andrew Howroyd_, Jan 28 2020