A020701 Pisot sequences E(3,5), P(3,5).

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

Number of meaningful differential operations of the (n+1)-th order on the space R^3. - Branko Malesevic, Feb 29 2004

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

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.

Branko Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.

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

Formula

a(n) = Fib(n+4). a(n) = a(n-1) + a(n-2).

a(n) = A020695(n+1). - R. J. Mathar, May 28 2008

G.f.: (3+2*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-7+3*sqrt(5))+(1+sqrt(5))^n*(7+3*sqrt(5))))/sqrt(5). - Colin Barker, Jun 05 2016

E.g.f.: (7*sqrt(5)*sinh(sqrt(5)*x/2) + 15*cosh(sqrt(5)*x/2))*exp(x/2)/5. - Ilya Gutkovskiy, Jun 05 2016

Example

Meaningful second-order differential operations appear in the form of five compositions as follows: 1. div grad f 2. curl curl F 3. grad div F 4. div curl F (=0) 5. curl grad f (=0)
Meaningful third-order differential operations appear in the form of eight compositions as follows: 1. grad div grad f 2. curl curl curl F 3. div grad div F 4. div curl curl F (=0) 5. div curl grad f (=0) 6. curl curl grad f (=0) 7. curl grad div F (=0) 8. grad div curl F (=0)

Mathematica

CoefficientList[Series[(-2 z - 3)/(z^2 + z - 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{1, 1}, {3, 5}, 40] (* Harvey P. Dale, Apr 22 2013 *)

Prog

(PARI) a(n)=fibonacci(n+4) \\ Charles R Greathouse IV, Jan 17 2012
(Magma) [Fibonacci(n-4): n in [8..80]]; // Vincenzo Librandi, Jul 12 2015

Crossrefs

Subsequence of A020695 and hence A000045. See A008776 for definitions of Pisot sequences.

Cf. A039834, A020695, A071679, A116183.

Sequence in context: A265069 A265070 A071679 * A024885 A180459 A133605

Adjacent sequences: A020698 A020699 A020700 * A020702 A020703 A020704

Keyword

nonn,easy

Author

David W. Wilson

Status

approved