A020695 Pisot sequence E(2,3).

2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

Pisano period lengths: A001175. - R. J. Mathar, Aug 10 2012

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876 (See Corollary 1 (x)).
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Subsequence of A000045. See A008776 for definitions of Pisot sequences.

See A000045 for the Fibonacci numbers.

Programs

GAP

A020695:=List([0..10^3], n->Fibonacci(n+3)); # Muniru A Asiru, Sep 05 2017

Magma

[Fibonacci(n+3): n in [0..50]]; // Vincenzo Librandi, Apr 23 2011

Mathematica

CoefficientList[Series[(-x - 2)/(x^2 + x - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{1,1},{2,3},40] (* or *) Fibonacci[Range[3,50]] (* Harvey P. Dale, Nov 22 2012 *)

PARI

a(n)=fibonacci(n+3) \\ Charles R Greathouse IV, Jan 17 2012

PARI

Vec((2+x)/(1-x-x^2) + O(x^40)) \\ Colin Barker, Jun 05 2016

Formula

a(n) = Fibonacci(n+3); a(n) = a(n-1) + a(n-2).

G.f.: (2+x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = (2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5))+(1+sqrt(5))^n*(2+sqrt(5))))/sqrt(5). - Colin Barker, Jun 05 2016

E.g.f.: 2*(2*sqrt(5)*sinh(sqrt(5)*x/2) + 5*cosh(sqrt(5)*x/2))*exp(x/2)/5. - Ilya Gutkovskiy, Jun 05 2016