A087124 -id:A087124 - cp's OEIS frontend

A087123 a(n) = Fibonacci(n+1) - (-1)^n*Fibonacci(n).

1, 2, 1, 5, 2, 13, 5, 34, 13, 89, 34, 233, 89, 610, 233, 1597, 610, 4181, 1597, 10946, 4181, 28657, 10946, 75025, 28657, 196418, 75025, 514229, 196418, 1346269, 514229, 3524578, 1346269, 9227465, 3524578, 24157817, 9227465, 63245986, 24157817

Offset: 0

Paul Barry, Aug 15 2003

Binomial transform is Fibonacci(n) + Fibonacci(2n+1) = A087124(n).

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000045, A001519, A001906, A087124.

with(combinat): A087123:=n->fibonacci(n+1)-(-1)^n*fibonacci(n): seq(A087123(n), n=0..50); # Wesley Ivan Hurt, Oct 05 2017

MapIndexed[#2 - (-1)^#1*#3 & @@ {First@ #2 - 1, Last@ #1, First@ #1} &, Partition[Fibonacci@ Range[0, 36], 2, 1]] (* or *)
CoefficientList[Series[(1 - x) (1 + 3 x + x^2)/((1 + x - x^2) (1 - x - x^2)), {x, 0, 38}], x] (* Michael De Vlieger, Oct 06 2017 *)

a(n) = fibonacci(n+1)-(-1)^n*fibonacci(n); \\ Altug Alkan, Oct 06 2017

a(2n) = Fibonacci(2n-1), a(2n+1) = Fibonacci(2n+3).
G.f.: (1-x)*(1+3*x+x^2)/((1+x-x^2)*(1-x-x^2)). - Colin Barker, Apr 16 2012
a(n) = 3*a(n-2) - a(n-4) for n > 3. - Wesley Ivan Hurt, Oct 05 2017

A192471 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^n+x^(2n+1).

Original entry on oeis.org

2, 5, 10, 24, 59, 150, 386, 1001, 2606, 6800, 17767, 46458, 121538, 318045, 832418, 2178920, 5703875, 14931950, 39090754, 102338337, 267921062, 701419680, 1836329615, 4807555634, 12586315394, 32951355125, 86267692666, 225851630136

Offset: 1

Clark Kimberling, Jul 01 2011

nonn

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first four polynomials p(n,x) and their reductions are as follows:
p(1,x)=1+x+x^3 -> 2+3x
p(2,x)=1+x^2+x^5 -> 5+6x
p(3,x)=1+x^3+x^7 -> 10+15x
p(4,x)=1+x^4+x^9 -> 24+37x.
From these, read
A192471=(2,5,10,24,...) and A087124=(3,6,15,37,...)

Cf. A192232.

Remove["Global`*"];
q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n+1);
Table[Simplify[p[n, x]], {n, 1, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192471 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A087124 *)

Empirical G.f.: -x*(2*x-1)*(x^3-3*x^2-x+2)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). [Colin Barker, Nov 12 2012]

cp's OEIS Frontend

A087123 a(n) = Fibonacci(n+1) - (-1)^n*Fibonacci(n).

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