A249450 a(n) = Fibonacci(2*n) - 2.
-2, -1, 1, 6, 19, 53, 142, 375, 985, 2582, 6763, 17709, 46366, 121391, 317809, 832038, 2178307, 5702885, 14930350, 39088167, 102334153, 267914294, 701408731, 1836311901, 4807526974, 12586269023, 32951280097, 86267571270, 225851433715, 591286729877
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (4,-4,1).
Programs
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Magma
[Fibonacci(2*n)-2: n in [0..40]];
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Mathematica
Table[Fibonacci[2 n] - 2, {n, 0, 40}] (* or *) CoefficientList[Series[(-2 + 7 x - 3 x^2) / (1 - 4 x + 4 x^2 - x^3), {x, 0, 40}], x] LinearRecurrence[{4, -4, 1}, {-2, -1, 1}, 30] (* Robert G. Wilson v, Dec 19 2014 *) -
PARI
Vec((-2+7*x-3*x^2)/(1-4*x+4*x^2-x^3) + O(x^30)) \\ Colin Barker, Nov 03 2016
Formula
G.f.: (-2+7*x-3*x^2)/(1-4*x+4*x^2-x^3).
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>2.
a(n) = 3*a(n-1) - a(n-2) + 2.
a(n) = (-2-((3-sqrt(5))/2)^n/sqrt(5)+((3+sqrt(5))/2)^n/sqrt(5)). - Colin Barker, Nov 03 2016
E.g.f.: 2*exp(3*x/2)*sinh(sqrt(5)*x/2)/sqr(5) - 2*exp(x). - Stefano Spezia, Jun 02 2024