A152718 a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) + a(n-5), n > 5.

1, 1, 1, 1, 2, 4, 8, 16, 31, 60, 117, 228, 444, 865, 1685, 3282, 6393, 12453, 24257, 47250, 92038, 179280, 349218, 680239, 1325032, 2581019, 5027546, 9793116, 19075931, 37157851, 72379476, 140987393, 274628197, 534946033, 1042017030, 2029736504, 3953707240

Offset: 0

Author

R. J. Mathar, Dec 11 2008

Keywords

• easy
• nonn

Comments

Conjecture: this is the inverse Motzkin transform of A054393.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..3435
• Index entries for linear recurrences with constant coefficients, signature (2, -1, 2, -1, 1).

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 + x*(-1+x+x^3)/(-1+2*x-x^2+2*x^3-x^4+x^5))); // G. C. Greubel, Sep 01 2018

Maple

a:= n-> `if`(n=0, 1, (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,
       <0|0|0|0|1>, <1|-1|2|-1|2>>^n. <<0, 1, 1, 1, 2>>)[1$2]):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 08 2016

Mathematica

Join[{1}, LinearRecurrence[{2, -1, 2, -1, 1}, {1, 1, 1, 2, 4}, 50]] (* G. C. Greubel, Sep 01 2018 *)

PARI

a(n) = {if (n==0, return(1)); if (n==1, return(1)); if (n==2, return(1)); if (n==3, return(1)); if (n==4, return(2)); if (n==5, return(4)); return (2*a(n-1)-a(n-2)+2*a(n-3)-a(n-4)+a(n-5));} \\ Michel Marcus, Jul 23 2013

PARI

x='x+O('x^50); Vec(1 + x*(-1+x+x^3)/(-1+2*x-x^2+2*x^3-x^4+x^5)) \\ G. C. Greubel, Sep 01 2018

Formula

a(n)=1, n < 4. a(4)=2, a(5)=4. a(n) = 2*a(n-1) - a(n-2) + 2a(n-3) - a(n-4) + a(n-5), n > 5.

G.f.: 1 + x*(-1 + x + x^3)/(-1 + 2*x - x^2 + 2*x^3 - x^4 + x^5). - R. J. Mathar, May 26 2016