A022113 Fibonacci sequence beginning 2, 7.

2, 7, 9, 16, 25, 41, 66, 107, 173, 280, 453, 733, 1186, 1919, 3105, 5024, 8129, 13153, 21282, 34435, 55717, 90152, 145869, 236021, 381890, 617911, 999801, 1617712, 2617513, 4235225, 6852738, 11087963, 17940701, 29028664, 46969365, 75998029, 122967394

Offset

0,1

References

H. S. M. Coxeter, Introduction to Geometry, Second Edition, Wiley Classics Library Edition Published 1989, p. 172.

Links

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

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

Formula

From Colin Barker, Oct 18 2013: (Start)

G.f.: -(5*x + 2)/(x^2 + x - 1).

a(n) = a(n-1) + a(n-2). (End)

a(n) = ((5+6*sqrt(5))/5)*((1+sqrt(5))/2)^n + ((5-6*sqrt(5))/5)*((1-sqrt(5))/2)^n starting at n=0. - Bogart B. Strauss, Oct 27 2013

a(n) = h*Fibonacci(n+k) + Fibonacci(n+k-h) with h=5, k=1. - Bruno Berselli, Feb 20 2017

a(n) = 8*F(n) + F(n-3) for F = A000045. - J. M. Bergot, Jul 14 2017

a(n) = Fibonacci(n+4) + Lucas(n-1). - Greg Dresden and Henry Sauer, Mar 04 2022

E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 6*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Jul 18 2022

Mathematica

RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == a[n - 1] + a[n - 2]}, a, {n, 0, 40}] (* Bruno Berselli, Mar 12 2015 *)
LinearRecurrence[{1, 1}, {2, 7}, 37] (* or *)
CoefficientList[Series[-(5 x + 2)/(x^2 + x - 1), {x, 0, 36}], x] (* Michael De Vlieger, Jul 14 2017 *)

Prog

(Magma) a0:=2; a1:=7; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013
(PARI) a(n)=8*fibonacci(n)+fibonacci(n-3) \\ Charles R Greathouse IV, Jul 14 2017
(PARI) a(n)=([0, 1; 1, 1]^n*[2; 7])[1, 1] \\ Charles R Greathouse IV, Jul 14 2017

Crossrefs

Cf. A000032. A000045.

Sequence in context: A165995 A287575 A267212 * A041643 A041395 A042345

Adjacent sequences: A022110 A022111 A022112 * A022114 A022115 A022116

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved