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A218694 Carlitz compositions of n into odd parts. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A057040 A096235 A147851 * A143596 A091712 A125721

Adjacent sequences:  A218691 A218692 A218693 * A218695 A218696 A218697

KEYWORD

nonn

AUTHOR

Joerg Arndt, Nov 04 2012

STATUS

approved

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Last modified November 18 18:46 EST 2017. Contains 294894 sequences.