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 A020701 Pisot sequences E(3,5), P(3,5). 12
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Number of meaningful differential operations of the k-th order on the space R^3. - Branko Malesevic, Feb 29 2004 Arxiv paper of 2007 generalizes Malesevic reference of 1998, giving recurrence relations through dimension 10, for which case f(i+6) = 5f(i+4) - 6f(i+2) + f(i). - Jonathan Vos Post, Apr 07 2007 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 B. 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. Sequence in context: A265069 A265070 A071679 * A024885 A180459 A133605 Adjacent sequences:  A020698 A020699 A020700 * A020702 A020703 A020704 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 19 08:16 EDT 2021. Contains 348074 sequences. (Running on oeis4.)