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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved