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A107428 Number of gap-free compositions of n. 4
1, 2, 4, 6, 11, 21, 39, 71, 141, 276, 542, 1070, 2110, 4189, 8351, 16618, 33134, 66129, 131937, 263483, 526453, 1051984, 2102582, 4203177, 8403116, 16800894, 33593742, 67174863, 134328816, 268624026, 537192064, 1074288649, 2148414285, 4296543181, 8592585289 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A gap-free composition contains all the parts between its smallest and largest part. a(5)=11 because we have: 5, 3+2, 2+3, 2+2+1, 2+1+2, 1+2+2, 2+1+1+1, 1+2+1+1, 1+1+2+1, 1+1+1+2, 1+1+1+1+1. - Geoffrey Critzer, Apr 13 2014

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Alois P. Heinz, Plot of (a(n)-2^(n-2))/2^(n-2) for n = 42..1000

P. Hitczenko and A. Knopfmacher, Gap-free compositions and gap-free samples of geometric random variables, Discrete Math., 294 (2005), 225-239.

FORMULA

a(n) ~ 2^(n-2). - Alois P. Heinz, Dec 07 2014

MAPLE

b:= proc(n, i, t) option remember; `if`(n=0, t!,

      `if`(i<1 or n<i, 0, add(b(n-i*j, i-1, t+j)/j!, j=1..n/i)))

    end:

a:= n-> add(b(n, i, 0), i=1..n):

seq(a(n), n=1..40);  # Alois P. Heinz, Apr 14 2014

MATHEMATICA

Table[Length[Select[Level[Map[Permutations, IntegerPartitions[n]], {2}], Length[Union[#]]==Max[#]-Min[#]+1&]], {n, 1, 20}] (* Geoffrey Critzer, Apr 13 2014 *)

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t!, If[i < 1 || n < i, 0, Sum[b[n - i*j, i - 1, t + j]/j!, {j, 1, n/i}]]]; a[n_] := Sum[b[n, i, 0], {i, 1, n}]; Table[a[n], {n, 1, 40}] (* Jean-Fran├žois Alcover, Aug 30 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A107429, A034296, A251729.

Sequence in context: A017993 A049870 A093970 * A086379 A096460 A322051

Adjacent sequences:  A107425 A107426 A107427 * A107429 A107430 A107431

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, May 26 2005

EXTENSIONS

More terms from Vladeta Jovovic, May 26 2005

STATUS

approved

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Last modified October 22 07:48 EDT 2019. Contains 328315 sequences. (Running on oeis4.)