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 A192068 a(n) = Fibonacci(2*n) - (n mod 2). 3

%I

%S 0,3,7,21,54,144,376,987,2583,6765,17710,46368,121392,317811,832039,

%T 2178309,5702886,14930352,39088168,102334155,267914295,701408733,

%U 1836311902,4807526976,12586269024,32951280099,86267571271,225851433717,591286729878

%N a(n) = Fibonacci(2*n) - (n mod 2).

%C Previous name was: 1-sequence of reduction of Lucas sequence by x^2 -> x+1.

%C See A192232 for definition of "k-sequence of reduction of [sequence] by [substitution]".

%F 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 a(n) = Fibonacci(2*n) - (n mod 2). - _Peter Luschny_, Mar 10 2015

%e (See A192243.)

%p a := n -> combinat[fibonacci](2*n)-(n mod 2):

%p seq(a(n), n=1..29); # _Peter Luschny_, Mar 10 2015

%t c[n_] := LucasL[n];

%t Table[c[n], {n, 1, 15}]

%t q[x_] := x + 1; p[0, x_] := 1;

%t p[n_, x_] := p[n - 1, x] + (x^n)*c[n + 1] reductionRules = {x^y_?EvenQ -> q[x]^(y/2),

%t x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,

%t 30}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192243 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192068 *)

%t (* by _Peter J. C. Moses_, Jun 26 2011 *)

%t Table[Fibonacci[2n]-Mod[n,2],{n,30}] (* _Harvey P. Dale_, Jul 11 2020 *)

%Y Cf. A000032, A000045, A192243, A192068.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jun 26 2011

%E New name from _Peter Luschny_, Mar 10 2015

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Last modified April 12 01:36 EDT 2021. Contains 342912 sequences. (Running on oeis4.)