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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000032. A000045.

Programs

Magma

a0:=2; a1:=7; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013

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 *)

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

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