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%I #27 Sep 08 2022 08:45:14
%S 0,1,1,2,1,3,4,3,7,10,7,17,24,17,41,58,41,99,140,99,239,338,239,577,
%T 816,577,1393,1970,1393,3363,4756,3363,8119,11482,8119,19601,27720,
%U 19601,47321,66922,47321,114243,161564,114243,275807,390050,275807,665857
%N a(n) = (a(n-1) mod 2)*a(n-1) + a(n-2) with a(0)=0, a(1)=1.
%C 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.
%H Colin Barker, <a href="/A097564/b097564.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Panario, M. Sahin, and Q. Wang, <a href="http://www.emis.de/journals/INTEGERS/papers/n78/n78.Abstract.html">A family of Fibonacci-like conditional sequences</a>, INTEGERS, Vol. 13, 2013, #A78.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,1).
%F From _Colin Barker_, Jun 01 2016: (Start)
%F a(n) = 2*a(n-3) + a(n-6) for n>5.
%F G.f.: x*(1+x+2*x^2-x^3+x^4) / (1-2*x^3-x^6). (End)
%p m:=50; S:=series( x*(1+x+2*x^2-x^3+x^4)/(1-2*x^3-x^6), x, m+1):
%p seq(coeff(S, x, j), j=0..m); # _G. C. Greubel_, Apr 20 2021
%t 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 *)
%o (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
%o (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
%o (Sage)
%o def A097564_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x*(1+x+2*x^2-x^3+x^4)/(1-2*x^3-x^6) ).list()
%o A097564_list(50) # _G. C. Greubel_, Apr 20 2021
%K nonn,easy
%O 0,4
%A _Gerald McGarvey_, Aug 27 2004
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