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A097895 Number of compositions of n with at least 1 odd and 1 even part. 1
0, 0, 2, 3, 11, 20, 51, 99, 222, 441, 935, 1872, 3863, 7751, 15774, 31653, 63939, 128232, 257963, 517011, 1037630, 2078417, 4165647, 8340192, 16702191, 33428943, 66912446, 133891725, 267921227, 536022488, 1072395555, 2145272571, 4291442718, 8584166169 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (3,1,-8,2,4).

FORMULA

G.f.: x^3*(3*x-2)/((2*x-1)*(2*x^2-1)*(x^2+x-1)). - Vladeta Jovovic, Sep 03 2004

a(n) = 3*a(n-1) + a(n-2) - 8*a(n-3) + 2*a(n-4) + 4*a(n-5) for n > 5. - Jinyuan Wang, Mar 10 2020

From Gregory L. Simay, May 27 2021: (Start)

a(2*n) = 2^(2*n - 1) - 2^(n-1) - A000045(2*n).

a(2*n+1) = 2^(2*n) - A000045(2*n + 1). (End)

EXAMPLE

n=4: 2+1+1, 1+2+1, 1+1+2. Total=3.

MAPLE

G:=x^3*(3*x-2)/((2*x-1)*(2*x^2-1)*(x^2+x-1)): Gser:=series(G, x=0, 37): seq(coeff(Gser, x^n), n=1..35); # Emeric Deutsch, Feb 15 2005

# second Maple program

b:= proc(n, o, e) option remember; `if`(n=0, `if`(o and e, 1, 0),

      add(`if`(irem(i, 2)=1, b(n-i, true, e),

                             b(n-i, o, true)), i=1..n))

    end:

a:= n-> b(n, false$2):

seq(a(n), n=1..50);  # Alois P. Heinz, Jun 11 2013

MATHEMATICA

e=(1-x^2)/(1-2x^2); o=(1-x^2)/(1-x-x^2); nn=30; Drop[CoefficientList[Series[(1-x)/(1-2x)-(o+e), {x, 0, nn}], x], 1]  (* Geoffrey Critzer, Jan 18 2012 *)

CROSSREFS

Cf. A000041 (partitions), A006477 (partitions of n with at least 1 odd and 1 even part), A000009 (partitions into odd parts), A035363 (partitions into even parts); A000079 (compositions). Compositions into odd parts give Fibonacci numbers (A000045), into even parts gives 0, 1, 0, 2, 0, 4, 0, 8, 0, 16, 0, 32, 0, 64, ... (essentially A000079).

Cf. A000045, A000041, A000009, A035363, A006477.

Cf. A007179.

Sequence in context: A086791 A291633 A004687 * A298351 A023182 A302310

Adjacent sequences:  A097892 A097893 A097894 * A097896 A097897 A097898

KEYWORD

nonn

AUTHOR

Dubois Marcel (dubois.ml(AT)club-internet.fr), Sep 03 2004

EXTENSIONS

More terms from Emeric Deutsch, Feb 15 2005

STATUS

approved

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Last modified January 26 09:35 EST 2022. Contains 350598 sequences. (Running on oeis4.)