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A054477 A Pellian-related sequence. 4
1, 13, 64, 307, 1471, 7048, 33769, 161797, 775216, 3714283, 17796199, 85266712, 408537361, 1957420093, 9378563104, 44935395427, 215298414031, 1031556674728, 4942484959609, 23680868123317, 113461855656976, 543628410161563, 2604680195150839, 12479772565592632 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jan 09 2016 *)

PROG

(Haskel)

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

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Last modified February 23 19:49 EST 2017. Contains 282507 sequences.