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