A174618 For n odd a(n) = a(n-2) + a(n-3), for n even a(n) = a(n-2) + a(n-5); with a(1) = 0, a(2) = 1.

0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 7, 10, 11, 17, 17, 28, 27, 45, 44, 72, 72, 116, 117, 188, 189, 305, 305, 494, 493, 799, 798, 1292, 1292, 2090, 2091, 3382, 3383, 5473, 5473, 8856, 8855, 14329, 14328, 23184, 23184, 37512, 37513, 60696

Offset: 1

Author

Mark Dols, Mar 23 2010

Keywords

• nonn
• easy

Comments

Combination a(2n)=A005252(n-1) and a(2n+1)=A024490(n). Consecutive pairs add up to A000045 and subtract to A010892. If a(1)= 1 formula gives: A103609.

Examples

			As consecutive pairs: (0,1),(0,1),(1,1),(2,1),(3,2),(4,4),...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1,0,0,0,1).

Crossrefs

Cf. A000045, A005252, A010892, A024490, A079977, A110161.

Programs

Magma

R:=PowerSeriesRing(Integers(), 70);
[0] cat Coefficients(R!( x^2*(1-x^2+x^3)/((1-x^2+x^4)*(1-x^2-x^4)) )); // G. C. Greubel, Oct 23 2024

Mathematica

nxt[{n_,a_,b_,c_,d_,e_}]:={n+1,b,c,d,e,If[EvenQ[n],d+c,d+a]}; NestList[nxt,{5,0,1,0,1,1},50][[All,2]] (* or *) LinearRecurrence[ {0,2,0,-1,0,0,0,1},{0,1,0,1,1,1,2,1},60] (* Harvey P. Dale, Nov 15 2019 *)

SageMath

def A174618(n): return (kronecker(12,n-3) - kronecker(12,n-2) + ((n+1)%2)*fibonacci(n//2) + (n%2)*fibonacci((n+1)//2))//2
[A174618(n) for n in range(1,71)] # G. C. Greubel, Oct 23 2024

Formula

G.f.: x^2*(1-x^2+x^3) / ( (1-x^2+x^4)*(1-x^2-x^4) ). - R. J. Mathar, Jan 27 2011

a(n) = (1/2)*(A110161(n-3) - A110161(n-2) + A079977(n-2) + A079977(n-1)). - G. C. Greubel, Oct 23 2024