cp's OEIS Frontend

A128695 Number of compositions of n with parts in N which avoid the adjacent pattern 111.

%I A128695 #49 Jul 09 2020 08:04:05
%S A128695 1,1,2,3,7,13,24,46,89,170,324,618,1183,2260,4318,8249,15765,30123,
%T A128695 57556,109973,210137,401525,767216,1465963,2801115,5352275,10226930,
%U A128695 19541236,37338699,71345449,136324309,260483548,497722578,951030367
%N A128695 Number of compositions of n with parts in N which avoid the adjacent pattern 111.
%H A128695 Alois P. Heinz, <a href="/A128695/b128695.txt">Table of n, a(n) for n = 0..1000</a>
%H A128695 S. Heubach and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0603285">Enumeration of 3-letter patterns in compositions</a>, arXiv:math/0603285 [math.CO], 2006
%H A128695 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%F A128695 G.f.: 1/(1-Sum(i>=1, x^i*(1+x^i)/(1+x^i*(1+x^i)) ) ).
%F A128695 a(n) ~ c * d^n, where d is the root of the equation Sum_{k>=1} 1/(d^k + 1/(1 + d^k)) = 1, d=1.9107639262818041675000243699745706859615884029961947632387839..., c=0.4993008137128378086219448701860326113802027003939127932922782... - _Vaclav Kotesovec_, May 01 2014, updated Jul 07 2020
%F A128695 For n>=2, a(n) = A091616(n) + A003242(n). - _Vaclav Kotesovec_, Jul 07 2020
%e A128695 From _Gus Wiseman_, Jul 06 2020: (Start)
%e A128695 The a(0) = 1 through a(5) = 13 compositions:
%e A128695   ()  (1)  (2)    (3)    (4)      (5)
%e A128695            (1,1)  (1,2)  (1,3)    (1,4)
%e A128695                   (2,1)  (2,2)    (2,3)
%e A128695                          (3,1)    (3,2)
%e A128695                          (1,1,2)  (4,1)
%e A128695                          (1,2,1)  (1,1,3)
%e A128695                          (2,1,1)  (1,2,2)
%e A128695                                   (1,3,1)
%e A128695                                   (2,1,2)
%e A128695                                   (2,2,1)
%e A128695                                   (3,1,1)
%e A128695                                   (1,1,2,1)
%e A128695                                   (1,2,1,1)
%e A128695 (End)
%p A128695 b:= proc(n, t) option remember; `if`(n=0, 1, add(`if`(abs(t)<>j,
%p A128695        b(n-j, j), `if`(t=-j, 0, b(n-j, -j))), j=1..n))
%p A128695     end:
%p A128695 a:= n-> b(n, 0):
%p A128695 seq(a(n), n=0..40);  # _Alois P. Heinz_, Nov 23 2013
%t A128695 nn=33;CoefficientList[Series[1/(1-Sum[(x^i+x^(2i))/(1+x^i+x^(2i)),{i,1,nn}]),{x,0,nn}],x] (* _Geoffrey Critzer_, Nov 23 2013 *)
%t A128695 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{___,x_,x_,x_,___}]&]],{n,13}] (* _Gus Wiseman_, Jul 06 2020 *)
%Y A128695 Column k=0 of A232435.
%Y A128695 Cf. A091616, A232432.
%Y A128695 The matching version is A335464.
%Y A128695 Contiguously (1,1)-avoiding compositions is A003242.
%Y A128695 Contiguously (1,1)-matching compositions are A261983.
%Y A128695 Compositions with some part > 2 are A008466
%Y A128695 Compositions by number of adjacent equal parts are A106356.
%Y A128695 Compositions where each part is adjacent to an equal part are A114901.
%Y A128695 Compositions with adjacent parts coprime are A167606.
%Y A128695 Compositions with equal parts contiguous are A274174.
%Y A128695 Patterns contiguously matched by compositions are A335457.
%Y A128695 Patterns contiguously matched by a given partition are A335516.
%Y A128695 Cf. A005251, A032020, A131044, A178470, A242882, A333175, A335455.
%K A128695 nonn
%O A128695 0,3
%A A128695 _Ralf Stephan_, May 08 2007