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

0,5

Comments

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

Links

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

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A147851 A367088 A321380 * A143596 A342763 A091712

Adjacent sequences: A218691 A218692 A218693 * A218695 A218696 A218697

Keyword

nonn

Author

Joerg Arndt, Nov 04 2012

Status

approved