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A192068 a(n) = Fibonacci(2*n) - (n mod 2).

%I A192068 #26 Jun 24 2022 22:56:06
%S A192068 0,3,7,21,54,144,376,987,2583,6765,17710,46368,121392,317811,832039,
%T A192068 2178309,5702886,14930352,39088168,102334155,267914295,701408733,
%U A192068 1836311902,4807526976,12586269024,32951280099,86267571271,225851433717,591286729878
%N A192068 a(n) = Fibonacci(2*n) - (n mod 2).
%C A192068 Previous name was: 1-sequence of reduction of Lucas sequence by x^2 -> x+1.
%C A192068 See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".
%F A192068 Empirical g.f. and recurrence: x^2*(3-2*x)/(1-3*x+3*x^3-x^4), a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - _Colin Barker_, Feb 08 2012
%F A192068 a(n) = Fibonacci(2*n) - (n mod 2). - _Peter Luschny_, Mar 10 2015
%e A192068 (See A192243.)
%p A192068 a := n -> combinat[fibonacci](2*n)-(n mod 2):
%p A192068 seq(a(n), n=1..29); # _Peter Luschny_, Mar 10 2015
%t A192068 c[n_] := LucasL[n];
%t A192068 Table[c[n], {n, 1, 15}]
%t A192068 q[x_] := x + 1; p[0, x_] := 1;
%t A192068 p[n_, x_] :=  p[n - 1, x] + (x^n)*c[n + 1] reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192068    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192068 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,
%t A192068    30}]
%t A192068 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192243 *)
%t A192068 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192068 *)
%t A192068 (* _Peter J. C. Moses_, Jun 26 2011 *)
%t A192068 Table[Fibonacci[2n]-Mod[n,2],{n,30}] (* _Harvey P. Dale_, Jul 11 2020 *)
%Y A192068 Cf. A000032, A000045, A192243, A192068.
%Y A192068 Partial sums of A081714.
%K A192068 nonn
%O A192068 1,2
%A A192068 _Clark Kimberling_, Jun 26 2011
%E A192068 New name from _Peter Luschny_, Mar 10 2015