A218694 Carlitz compositions of n into odd parts.

1, 1, 0, 1, 2, 2, 2, 3, 6, 9, 10, 13, 22, 32, 40, 56, 86, 122, 164, 229, 332, 474, 656, 914, 1310, 1867, 2604, 3648, 5184, 7346, 10318, 14506, 20516, 29022, 40880, 57548, 81260, 114810, 161864, 228092, 321892, 454444, 640954, 903715, 1274998, 1799320, 2538218, 3579714, 5049954, 7125359, 10051844

Offset: 0

Author

Joerg Arndt, Nov 04 2012

Keywords

• nonn

Comments

Carlitz compositions are compositions where adjacent parts are distinct (see A003242).

Examples

			There are a(12) = 22 such compositions of 12:
[ 1]  1 3 1 3 1 3
[ 2]  1 3 1 7
[ 3]  1 3 5 3
[ 4]  1 3 7 1
[ 5]  1 5 1 5
[ 6]  1 7 1 3
[ 7]  1 7 3 1
[ 8]  1 11
[ 9]  3 1 3 1 3 1
[10]  3 1 3 5
[11]  3 1 5 3
[12]  3 1 7 1
[13]  3 5 1 3
[14]  3 5 3 1
[15]  3 9
[16]  5 1 5 1
[17]  5 3 1 3
[18]  5 7
[19]  7 1 3 1
[20]  7 5
[21]  9 3
[22]  11 1
		

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms 0..262 from Joerg Arndt)

Crossrefs

Cf. A003242 (Carlitz compositions), A032021 (compositions into distinct odd parts), A032020 (compositions into distinct parts).

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1,
       add(`if`(j=t or irem(j, 2)=0, 0, b(n-j, j)), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Nov 08 2012

Mathematica

nn=20;CoefficientList[Series[1/(1-Sum[z^(2j+1)/(1+z^(2j+1)),{j,0,nn}]),{z,0,nn}],z] (* Geoffrey Critzer, Nov 21 2013 *)

Formula

G.f.: 1/( 1 - Sum_{j>=0} x^(2j+1)/(1 + x^(2j+1)) ). - Geoffrey Critzer, Nov 21 2013

a(n) ~ c / r^n, where r = 0.708865489663179258570259601255070249415... is the root of the equation sum_{j>=0} x^(2j+1)/(1 + x^(2j+1)) = 1, c = 0.3391570949344217123793275284038135702369824934927187... . - Vaclav Kotesovec, Aug 22 2014