OFFSET
0,3
COMMENTS
From L. Edson Jeffery, Apr 19 2011: (Start)
Let A be the unit-primitive matrix (see [Jeffery])
A = A_(10,4) =
(0 0 0 0 1)
(0 0 0 2 0)
(0 0 2 0 1)
(0 2 0 2 0)
(2 0 2 0 1).
Then a(n) = (1/5)*trace(A^n). (End)
a(n-1)+1 is the number of paths to reach a position outside a 4 X 4 chessboard after n steps, starting in one of the corners, when performing a walk with unit steps on the square lattice. - Ruediger Jehn, Oct 10 2024
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 875
L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (3,2,-4).
FORMULA
Recurrence: {a(1)=1, a(0)=1, -4*a(n) - 2*a(n+1) + a(n+2) + 1 = 0}.
a(n) = Sum((-1/25)*(-1-8*_alpha+4*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z-2*_Z^2+4*_Z^3)).
a(n)/a(n-1) tends to (1 + sqrt(5)) = 3.236067... - Gary W. Adamson, Mar 01 2008
a(n) = (1/5) * Sum_{k=1..5} ((x_k)^4-3*(x_k)^2+1), x_k=2*cos((2*k-1)*Pi/10). Also, a(n)/a(n-1) -> spectral radius of matrix A_(10,4) above. - L. Edson Jeffery, Apr 19 2011
a(n) = (2*A087131(n)+1)/5. - Bruno Berselli, Apr 20 2011
a(n) = (2/5)*((1+sqrt(5))^n + (1-sqrt(5))^n + 1/2). - Ruediger Jehn, Sep 29 2024
E.g.f.: exp(x)*(1 + 4*cosh(sqrt(5)*x))/5. - Stefano Spezia, Oct 02 2024
MAPLE
spec := [S, {S=Sequence(Prod(Union(Sequence(Union(Z, Z)), Z, Z), Z))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
CoefficientList[Series[(1-2x)/((x-1)(4x^2+2x-1)), {x, 0, 40}], x] (* or *) LinearRecurrence[{3, 2, -4}, {1, 1, 5}, 40] (* Harvey P. Dale, Jul 10 2017 *)
PROG
(Sage) from sage.combinat.sloane_functions import recur_gen2b
it = recur_gen2b(1, 1, 2, 4, lambda n:-1)
[next(it) for i in range(1, 28)] # Zerinvary Lajos, Jul 09 2008
(Magma) [(1/5)*(2^(n+1)*Lucas(n)+1): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011
(Maxima) makelist(coeff(taylor((1-2*x)/(1-3*x-2*x^2+4*x^3), x, 0, n), x, n), n, 0, 25); /* Bruno Berselli, May 30 2011 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 08 2000
STATUS
approved