|
|
A128695
|
|
Number of compositions of n with parts in N which avoid the adjacent pattern 111.
|
|
20
|
|
|
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
|
|
|
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
|
|
EXAMPLE
|
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):
|
|
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
|
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.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|