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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
A. F. Horadam, Pell Identities, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
Tanya Khovanova, Recursive Sequences
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
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Apr 16 2000
STATUS
approved

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Last modified April 16 22:11 EDT 2024. Contains 371755 sequences. (Running on oeis4.)