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 A373952 #13 Jul 01 2024 20:10:51
%S A373952 0,0,0,3,2,4,5,6,6,9,8,10,11,12,12,15,14,16,17,18,18,21,20,22,23,24,
%T A373952 24,27,26,28,29,30,30,33,32,34,35,36,36,39,38,40,41,42,42,45,44,46,47,
%U A373952 48,48,51,50,52,53,54,54,57,56,58,59,60,60,63,62,64,65,66
%N A373952 Number of integer compositions of n whose run-compression sums to 3.
%C A373952 We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).
%H A373952 John Tyler Rascoe, <a href="/A373952/b373952.txt">Table of n, a(n) for n = 0..10000</a>
%F A373952 G.f.: x^3 * (3-x-x^2-x^3)/((1-x)*(1-x^2)*(1-x^3)). - _John Tyler Rascoe_, Jul 01 2024
%e A373952 The a(3) = 3 through a(9) = 9 compositions:
%e A373952 (3) (112) (122) (33) (1222) (11222) (333)
%e A373952 (12) (211) (221) (1122) (2221) (22211) (12222)
%e A373952 (21) (1112) (2211) (11122) (111122) (22221)
%e A373952 (2111) (11112) (22111) (221111) (111222)
%e A373952 (21111) (111112) (1111112) (222111)
%e A373952 (211111) (2111111) (1111122)
%e A373952 (2211111)
%e A373952 (11111112)
%e A373952 (21111111)
%t A373952 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#]]==3&]],{n,0,10}]
%o A373952 (PARI)
%o A373952 A_x(N)={my(x='x+O('x^N)); concat([0, 0, 0], Vec(x^3 *(3-x-x^2-x^3)/((1-x)*(1-x^2)*(1-x^3))))}
%o A373952 A_x(50) \\ _John Tyler Rascoe_, Jul 01 2024
%Y A373952 For partitions we appear to have A137719.
%Y A373952 Column k = 3 of A373949, rows-reversed A373951.
%Y A373952 The compression-sum statistic is represented by A373953, difference A373954.
%Y A373952 A003242 counts compressed compositions (anti-runs).
%Y A373952 A011782 counts compositions.
%Y A373952 A114901 counts compositions with no isolated parts.
%Y A373952 A116861 counts partitions by compressed sum, by compressed length A116608.
%Y A373952 A124767 counts runs in standard compositions, anti-runs A333381.
%Y A373952 A240085 counts compositions with no unique parts.
%Y A373952 A333755 counts compositions by compressed length.
%Y A373952 A373948 represents the run-compression transformation.
%Y A373952 Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333489, A373950.
%K A373952 nonn
%O A373952 0,4
%A A373952 _Gus Wiseman_, Jun 29 2024
%E A373952 a(26) onwards from _John Tyler Rascoe_, Jul 01 2024