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A114901 Number of compositions of n such that each part is adjacent to an equal part. 46
1, 0, 1, 1, 2, 1, 5, 3, 10, 10, 21, 22, 49, 51, 105, 126, 233, 292, 529, 678, 1181, 1585, 2654, 3654, 6016, 8416, 13606, 19395, 30840, 44517, 70087, 102070, 159304, 233941, 362429, 535520, 825358, 1225117, 1880220, 2801749, 4285086, 6404354, 9769782, 14634907 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
N. J. A. Sloane, Transforms
FORMULA
INVERT(iMOEBIUS(iINVERT(A000012 shifted right 2 places)))
EXAMPLE
The 5 compositions of 6 are 3+3, 2+2+2, 2+2+1+1, 1+1+2+2, 1+1+1+1+1+1.
From Gus Wiseman, Nov 25 2019: (Start)
The a(2) = 1 through a(9) = 10 compositions:
(11) (111) (22) (11111) (33) (11122) (44) (333)
(1111) (222) (22111) (1133) (11133)
(1122) (1111111) (2222) (33111)
(2211) (3311) (111222)
(111111) (11222) (222111)
(22211) (1111122)
(111122) (1112211)
(112211) (1122111)
(221111) (2211111)
(11111111) (111111111)
(End)
MAPLE
g:= proc(n, i) option remember; add(b(n-i*j, i), j=2..n/i) end:
b:= proc(n, l) option remember; `if`(n=0, 1,
add(`if`(i=l, 0, g(n, i)), i=1..n/2))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..50); # Alois P. Heinz, Nov 29 2019
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Min@@Length/@Split[#]>1&]], {n, 0, 10}] (* Gus Wiseman, Nov 25 2019 *)
g[n_, i_] := g[n, i] = Sum[b[n - i*j, i], {j, 2, n/i}] ;
b[n_, l_] := b[n, l] = If[n==0, 1, Sum[If[i==l, 0, g[n, i]], {i, 1, n/2}]];
a[n_] := b[n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *)
CROSSREFS
The case of partitions is A007690.
Compositions with no adjacent parts equal are A003242.
Compositions with all multiplicities > 1 are A240085.
Compositions with minimum multiplicity 1 are A244164.
Compositions with at least two adjacent parts equal are A261983.
Sequence in context: A085261 A179218 A131119 * A355562 A194809 A113178
KEYWORD
nonn
AUTHOR
Christian G. Bower, Jan 05 2006
STATUS
approved

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Last modified April 22 13:50 EDT 2024. Contains 371903 sequences. (Running on oeis4.)