A188577 -id:A188577 - cp's OEIS frontend

A188575 Number of non-complete compositions of n.

0, 1, 1, 4, 8, 14, 31, 63, 129, 248, 509, 1011, 2044, 4089, 8167, 16360, 32725, 65482, 131017, 262176, 524167, 1048678, 2096985, 4194358, 8387802, 16776408, 33550943, 67101615, 134199983, 268399122, 536793004, 1073590077, 2147187353, 4294419287, 8588940438

Offset: 1

N. J. A. Sloane, Apr 04 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Archibald and A. Knopfmacher, The largest missing value in a composition of an integer, Discrete Math., 311 (2011), 723-731.
Alois P. Heinz, Plot of (a(n)-2^(n-2))/2^(n-2) for n = 60..1000

Cf. A107429, A188576, A188577, A251729.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, t!, 0),
      `if`(i<1 or n 2^(n-1) -add(b(n, i, 0), i=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 06 2014

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[i == 0, t!, 0], If[i<1 || nJean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

a(n) = 2^(n-1) - A107429(n) ~ 2^(n-2). - Alois P. Heinz, Dec 06 2014

More terms from Alois P. Heinz, Dec 06 2014

A188576 Number of compositions of n with the LMV property.

Original entry on oeis.org

0, 1, 1, 4, 6, 14, 26, 54, 105, 213, 423, 849, 1697, 3399, 6799, 13608, 27220, 54451, 108901, 217789, 435517, 870892, 1741467, 3482322, 6963512, 13925078, 27846979, 55689150, 111371677, 222735709, 445466058, 890938357, 1781916885, 3563957177, 7128223846

Offset: 1

N. J. A. Sloane, Apr 04 2011

nonn

A composition has the largest missing value (LMV) property if its largest part is at least 2 and it does not contain a part one less than its largest part. - Andrew Howroyd, May 18 2020

			The a(5) = 6 compositions of 5 with the LMV property are: 5, 14, 41, 113, 131, 311. - _Andrew Howroyd_, May 18 2020

Andrew Howroyd, Table of n, a(n) for n = 1..1000
M. Archibald and A. Knopfmacher, The largest missing value in a composition of an integer, Discrete Math., 311 (2011), 723-731.

Cf. A188575, A188577.

seq(n)={Vec(sum(k=2, n, 1/(1 - x*(1-x^(k-2))/(1-x) - x^k) - 1/(1 - x*(1-x^(k-2))/(1-x))  + O(x*x^n)), -n)} \\ Andrew Howroyd, May 18 2020

G.f.: Sum_{k>=2} 1/(1 - x*(1-x^(k-2))/(1-x) - x^k) - 1/(1 - x*(1-x^(k-2))/(1-x)). - Andrew Howroyd, May 18 2020

Terms a(21) and beyond from Andrew Howroyd, May 18 2020

cp's OEIS Frontend

A188575 Number of non-complete compositions of n.

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