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

0,2

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

Cf. A002320.

Sequence in context: A092653 A067465 A166605 * A169883 A220564 A264513

Adjacent sequences: A054474 A054475 A054476 * A054478 A054479 A054480

Keyword

easy,nonn

Author

Barry E. Williams, Apr 16 2000

Status

approved