A097564 a(n) = (a(n-1) mod 2)*a(n-1) + a(n-2) with a(0)=0, a(1)=1.

0, 1, 1, 2, 1, 3, 4, 3, 7, 10, 7, 17, 24, 17, 41, 58, 41, 99, 140, 99, 239, 338, 239, 577, 816, 577, 1393, 1970, 1393, 3363, 4756, 3363, 8119, 11482, 8119, 19601, 27720, 19601, 47321, 66922, 47321, 114243, 161564, 114243, 275807, 390050, 275807, 665857

Offset: 0

Author

Gerald McGarvey, Aug 27 2004

Keywords

• nonn
• easy

Comments

The sequences a(2), a(5), ... a(1+3*n) ... and a(4), a(7), ... a(4 + 3n) ... are both A001333 (numerators of continued fraction convergents to sqrt(2)). The sequence a(0), a(3), a(6), ... a(3+3*n) ... is twice A000129 (the Pell nos. or the denominators of continued fraction convergents to sqrt(2).), also is A052542 starting w/ offset 1.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• D. Panario, M. Sahin, and Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,1).

Programs

Magma

[n le 2 select n-1 else (Self(n-1) mod 2)*Self(n-1)+Self(n-2): n in [1..50]]; // Bruno Berselli, Jun 02 2016

Maple

m:=50; S:=series( x*(1+x+2*x^2-x^3+x^4)/(1-2*x^3-x^6), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Apr 20 2021

Mathematica

nxt[{a_,b_}]:={b,Mod[b,2]*b+a}; NestList[nxt,{0,1},50][[All,1]] (* or *) LinearRecurrence[{0,0,2,0,0,1},{0,1,1,2,1,3},50] (* Harvey P. Dale, Aug 15 2017 *)

PARI

concat(0, Vec(x*(1+x+2*x^2-x^3+x^4)/(1-2*x^3-x^6) + O(x^100))) \\ Colin Barker, Jun 02 2016

Sage

def A097564_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x+2*x^2-x^3+x^4)/(1-2*x^3-x^6) ).list()
A097564_list(50) # G. C. Greubel, Apr 20 2021

Formula

From Colin Barker, Jun 01 2016: (Start)

a(n) = 2*a(n-3) + a(n-6) for n>5.

G.f.: x*(1+x+2*x^2-x^3+x^4) / (1-2*x^3-x^6). (End)