cp's OEIS Frontend

A335464 Number of compositions of n with a run of length > 2.

%I A335464 #13 May 21 2021 04:16:45
%S A335464 0,0,0,1,1,3,8,18,39,86,188,406,865,1836,3874,8135,17003,35413,73516,
%T A335464 152171,314151,647051,1329936,2728341,5587493,11424941,23327502,
%U A335464 47567628,96879029,197090007,400546603,813258276,1649761070,3343936929,6772740076,13707639491
%N A335464 Number of compositions of n with a run of length > 2.
%C A335464 A composition of n is a finite sequence of positive integers summing to n.
%C A335464 Also compositions contiguously matching the pattern (1,1,1).
%F A335464 a(n) = A011782(n) - A128695(n). - _Alois P. Heinz_, Jul 06 2020
%e A335464 The a(3) = 1 through a(7) = 18 compositions:
%e A335464   (111)  (1111)  (1112)   (222)     (1114)
%e A335464                  (2111)   (1113)    (1222)
%e A335464                  (11111)  (3111)    (2221)
%e A335464                           (11112)   (4111)
%e A335464                           (11121)   (11113)
%e A335464                           (12111)   (11122)
%e A335464                           (21111)   (11131)
%e A335464                           (111111)  (13111)
%e A335464                                     (21112)
%e A335464                                     (22111)
%e A335464                                     (31111)
%e A335464                                     (111112)
%e A335464                                     (111121)
%e A335464                                     (111211)
%e A335464                                     (112111)
%e A335464                                     (121111)
%e A335464                                     (211111)
%e A335464                                     (1111111)
%p A335464 b:= proc(n, t) option remember; `if`(n=0, 1, add(`if`(abs(t)<>j,
%p A335464        b(n-j, j), `if`(t=-j, 0, b(n-j, -j))), j=1..n))
%p A335464     end:
%p A335464 a:= n-> ceil(2^(n-1))-b(n, 0):
%p A335464 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 06 2020
%t A335464 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{___,x_,x_,x_,___}]&]],{n,0,10}]
%t A335464 (* Second program: *)
%t A335464 b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[If[Abs[t] != j,
%t A335464      b[n - j, j], If[t == -j, 0, b[n - j, -j]]], {j, 1, n}]];
%t A335464 a[n_] := Ceiling[2^(n-1)] - b[n, 0];
%t A335464 a /@ Range[0, 40] (* _Jean-François Alcover_, May 21 2021, after _Alois P. Heinz_ *)
%Y A335464 Compositions contiguously avoiding (1,1) are A003242.
%Y A335464 Compositions with some part > 2 are A008466.
%Y A335464 Compositions by number of adjacent equal parts are A106356.
%Y A335464 Compositions where each part is adjacent to an equal part are A114901.
%Y A335464 Compositions contiguously avoiding (1,1,1) are A128695.
%Y A335464 Compositions with adjacent parts coprime are A167606.
%Y A335464 Compositions contiguously matching (1,1) are A261983.
%Y A335464 Compositions with all equal parts contiguous are A274174.
%Y A335464 Patterns contiguously matched by compositions are A335457.
%Y A335464 Cf. A005251, A011782, A056986, A032020, A131044, A178470, A242882, A335455, A335458.
%K A335464 nonn
%O A335464 0,6
%A A335464 _Gus Wiseman_, Jul 06 2020
%E A335464 a(23)-a(35) from _Alois P. Heinz_, Jul 06 2020