A054477 A Pellian-related sequence.

1, 13, 64, 307, 1471, 7048, 33769, 161797, 775216, 3714283, 17796199, 85266712, 408537361, 1957420093, 9378563104, 44935395427, 215298414031, 1031556674728, 4942484959609, 23680868123317, 113461855656976, 543628410161563, 2604680195150839, 12479772565592632

Offset: 0

Author

Barry E. Williams, Apr 16 2000

Keywords

• easy
• nonn

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)

Crossrefs

Cf. A002320.

Programs

Haskell

a054477 n = a054477_list !! n
a054477_list = 1 : 13 :
   (zipWith (-) (map (* 5) (tail a054477_list)) a054477_list)
-- Reinhard Zumkeller, Oct 16 2011

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 *)

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