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A097564
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a(n) = (a(n-1) mod 2)*a(n-1) + a(n-2) with a(0)=0, a(1)=1.
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1
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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
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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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)
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
(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
(Sage)
def A097564_list(prec):
P.<x> = 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
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CROSSREFS
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Sequence in context: A133310 A077608 A002124 * A345233 A128270 A151550
Adjacent sequences: A097561 A097562 A097563 * A097565 A097566 A097567
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KEYWORD
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nonn,easy
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AUTHOR
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Gerald McGarvey, Aug 27 2004
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STATUS
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approved
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