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

0,1

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).

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

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 *)

Prog

(Magma) [Fibonacci(n+3): n in [0..50]]; // Vincenzo Librandi, Apr 23 2011
(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
(GAP)
A020695:=List([0..10^3], n->Fibonacci(n+3)); # Muniru A Asiru, Sep 05 2017

Crossrefs

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

See A000045 for the Fibonacci numbers.

Sequence in context: A333378 A000045 A324969 * A212804 A132916 A274163

Adjacent sequences: A020692 A020693 A020694 * A020696 A020697 A020698

Keyword

nonn,easy

Author

David W. Wilson

Status

approved