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 A128695 Number of compositions of n with parts in N which avoid the adjacent pattern 111. 9
 1, 1, 2, 3, 7, 13, 24, 46, 89, 170, 324, 618, 1183, 2260, 4318, 8249, 15765, 30123, 57556, 109973, 210137, 401525, 767216, 1465963, 2801115, 5352275, 10226930, 19541236, 37338699, 71345449, 136324309, 260483548, 497722578, 951030367 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 S. Heubach and T. Mansour, Enumeration of 3-letter patterns in compositions, arXiv:math/0603285 [math.CO], 2006 FORMULA G.f.: 1/(1-Sum(i>=1, x^i*(1+x^i)/(1+x^i*(1+x^i)) ) ). 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 For n>=2, a(n) = A091616(n) + A003242(n). - Vaclav Kotesovec, Jul 07 2020 EXAMPLE From Gus Wiseman, Jul 06 2020: (Start) The a(0) = 1 through a(5) = 13 compositions:   ()  (1)  (2)    (3)    (4)      (5)            (1,1)  (1,2)  (1,3)    (1,4)                   (2,1)  (2,2)    (2,3)                          (3,1)    (3,2)                          (1,1,2)  (4,1)                          (1,2,1)  (1,1,3)                          (2,1,1)  (1,2,2)                                   (1,3,1)                                   (2,1,2)                                   (2,2,1)                                   (3,1,1)                                   (1,1,2,1)                                   (1,2,1,1) (End) MAPLE b:= proc(n, t) option remember; `if`(n=0, 1, add(`if`(abs(t)<>j,        b(n-j, j), `if`(t=-j, 0, b(n-j, -j))), j=1..n))     end: a:= n-> b(n, 0): seq(a(n), n=0..40);  # Alois P. Heinz, Nov 23 2013 MATHEMATICA 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 *) Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#, {___, x_, x_, x_, ___}]&]], {n, 13}] (* Gus Wiseman, Jul 06 2020 *) CROSSREFS Column k=0 of A232435. Cf. A091616, A232432. The matching version is A335464. Contiguously (1,1)-avoiding compositions is A003242. Contiguously (1,1)-matching compositions are A261983. Compositions with some part > 2 are A008466 Compositions by number of adjacent equal parts are A106356. Compositions where each part is adjacent to an equal part are A114901. Compositions with adjacent parts coprime are A167606. Compositions with equal parts contiguous are A274174. Patterns contiguously matched by compositions are A335457. Patterns contiguously matched by a given partition are A335516. Cf. A005251, A032020, A131044, A178470, A242882, A333175, A335455. Sequence in context: A075058 A213968 A213967 * A024504 A256494 A088172 Adjacent sequences:  A128692 A128693 A128694 * A128696 A128697 A128698 KEYWORD nonn AUTHOR Ralf Stephan, May 08 2007 STATUS approved

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Last modified January 17 19:19 EST 2021. Contains 340247 sequences. (Running on oeis4.)