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A079501 Number of compositions of the integer n with strictly smallest part in the first position. 8
1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, 173, 275, 436, 695, 1107, 1769, 2831, 4537, 7276, 11683, 18774, 30194, 48592, 78247, 126062, 203192, 327645, 528518, 852815, 1376491, 2222294, 3588628, 5796196, 9363458, 15128631, 24447014, 39510108 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Also number of compositions of n such that the first part is divisible by the number of parts . [Vladeta Jovovic, Dec 02 2009]

REFERENCES

Arnold Knopfmacher and Neville Robbins, Compositions with parts constrained by the leading summand, Ars Combin. 76 (2005), 287-295.

LINKS

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

FORMULA

G.f.: sum(k>=1, (1-z)*z^k/(1-z-z^(k+1)) ).

Also sum(k>=1, z^(2*k-1)/((1-z^k)*(1-z)^(k-1)) ), cf. A105039. - Vladeta Jovovic, Apr 05 2005

a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n-2). - Vaclav Kotesovec, May 01 2014

EXAMPLE

The a(9)=19 such compositions of 9 are

[ 1]  [ 1 2 2 2 2 ]

[ 2]  [ 1 2 2 4 ]

[ 3]  [ 1 2 3 3 ]

[ 4]  [ 1 2 4 2 ]

[ 5]  [ 1 2 6 ]

[ 6]  [ 1 3 2 3 ]

[ 7]  [ 1 3 3 2 ]

[ 8]  [ 1 3 5 ]

[ 9]  [ 1 4 2 2 ]

[10]  [ 1 4 4 ]

[11]  [ 1 5 3 ]

[12]  [ 1 6 2 ]

[13]  [ 1 8 ]

[14]  [ 2 3 4 ]

[15]  [ 2 4 3 ]

[16]  [ 2 7 ]

[17]  [ 3 6 ]

[18]  [ 4 5 ]

[19]  [ 9 ]

- Joerg Arndt, Jan 01 2013

MAPLE

b:= proc(n, s) option remember; `if`(n=0, 1, add(

      `if`(n-j>0 and n-j<=s, 0, b(n-j, s)), j=s+1..n))

    end:

a:= n-> 1 +add(b(n-j, j), j=1..n/2):

seq(a(n), n=1..60);  # Alois P. Heinz, Apr 29 2014

MATHEMATICA

b[n_, s_] := b[n, s] = If[n == 0, 1, Sum[ If[n - j > 0 && n - j <= s, 0, b[n - j, s]], {j, s + 1, n}]]; a[n_] := 1 + Sum[b[n - j, j], {j, 1, n/2}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A168655, A168656, A168657. [From Vladeta Jovovic, Dec 02 2009]

Sequence in context: A172128 A274154 A274153 * A093335 A093333 A116085

Adjacent sequences:  A079498 A079499 A079500 * A079502 A079503 A079504

KEYWORD

nonn

AUTHOR

Arnold Knopfmacher, Jan 21 2003

EXTENSIONS

More terms from Benoit Cloitre, Jan 21 2003

STATUS

approved

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Last modified May 28 21:38 EDT 2017. Contains 287241 sequences.