A097895 Number of compositions of n with at least 1 odd and 1 even part.

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

Offset: 1

Author

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

Keywords

• nonn

Examples

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

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.

Programs

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 *)

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)

Extensions

More terms from Emeric Deutsch, Feb 15 2005