OFFSET
0,2
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 256.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (5,-1)
FORMULA
a(n) = 5a(n-1)-a(n-2); a(0)=1, a(1)=13.
(A054477)=sqrt{21*(A002320)^2-20}; where the algebraic operations on (A------) are performed from the inside - out; that is, first squared, then multiplied by 21, then 20 is subtracted and finally the square root is performed term by term.
G.f.: (1+8*x)/(1-5*x+x^2). - Alois P. Heinz, Aug 07 2008
a(n) = (2^(-1-n)*((5-sqrt(21))^n*(-21+sqrt(21))+(5+sqrt(21))^n*(21+sqrt(21))))/sqrt(21). - Colin Barker, May 26 2016
E.g.f.: (sqrt(21)*sinh(sqrt(21)*x/2) + cosh(sqrt(21)*x/2))*exp(5*x/2). - Ilya Gutkovskiy, May 26 2016
MAPLE
a:= n-> (Matrix([[1, -8]]). Matrix([[5, 1], [ -1, 0]])^(n))[1, 1]:
seq(a(n), n=0..20); # Alois P. Heinz, Aug 07 2008
MATHEMATICA
LinearRecurrence[{5, -1}, {1, 13}, 20] (* Jean-François Alcover, Jan 09 2016 *)
PROG
(Haskell)
a054477 n = a054477_list !! n
a054477_list = 1 : 13 :
(zipWith (-) (map (* 5) (tail a054477_list)) a054477_list)
-- Reinhard Zumkeller, Oct 16 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Apr 16 2000
STATUS
approved