A114901 Number of compositions of n such that each part is adjacent to an equal part.

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

Offset: 0

Author

Christian G. Bower, Jan 05 2006

Keywords

• nonn

Examples

			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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4000
• N. J. A. Sloane, Transforms

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.

Cf. A178470, A238130, A274174, A329863.

Programs

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 *)

PARI

A_x(N,k) = { my(x='x+O('x^N), g=1/(1-sum(i=1,N,sum(j=k+1,N, x^(i*j))/(1+ sum(j=k+1,N, x^(i*j)))))); Vec(g)}
A_x(50,1) \\ John Tyler Rascoe, May 17 2024

Formula

INVERT(iMOEBIUS(iINVERT(A000012 shifted right 2 places)))

G.f.: A(x,1) is the k = 1 case of A(x,k) = 1/(1 - Sum_{i>0} ( (Sum_{j>k} x^(i*j))/(1 + Sum_{j>k} x^(i*j)) )) where A(x,k) is the g.f. for compositions of n with all run-lengths > k. - John Tyler Rascoe, May 16 2024