A097136 a(n) = 3*Fibonacci(2*n) + 1.

1, 4, 10, 25, 64, 166, 433, 1132, 2962, 7753, 20296, 53134, 139105, 364180, 953434, 2496121, 6534928, 17108662, 44791057, 117264508, 307002466, 803742889, 2104226200, 5508935710, 14422580929, 37758807076, 98853840298, 258802713817, 677554301152, 1773860189638

Offset

0,2

Comments

Binomial transform of A097135.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Formula

G.f.: (1-2*x^2) / ((1-x)*(1-3*x+x^2)).

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).

a(n) = 1 + 3*((3+sqrt(5))/2)^n/sqrt(5) - 3*((3-sqrt(5))/2)^n/sqrt(5).

a(n) = A097132(2*n) = A097133(2*n).

E.g.f.: cosh(x) + sinh(x) + 6*exp(3*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Sep 29 2025

Mathematica

Table[3*Fibonacci[2n]+1, {n, 0, 30}] (* or *) LinearRecurrence[{4, -4, 1}, {1, 4, 10}, 30] (* Harvey P. Dale, May 25 2018 *)

Prog

(PARI) Vec((1-2*x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016

Crossrefs

Cf. A000045, A097132, A097133, A097135.

Sequence in context: A298412 A289245 A347725 * A049348 A282389 A394943

Adjacent sequences: A097133 A097134 A097135 * A097137 A097138 A097139

Keyword

nonn,easy

Author

Paul Barry, Jul 26 2004

Status

approved