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A081002 a(n) = Fibonacci(4n) + 1, or Fibonacci(2n-1)*Lucas(2n+1). 1
1, 4, 22, 145, 988, 6766, 46369, 317812, 2178310, 14930353, 102334156, 701408734, 4807526977, 32951280100, 225851433718, 1548008755921, 10610209857724, 72723460248142, 498454011879265, 3416454622906708, 23416728348467686 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

If x=a(n), y=a(n+1), z=a(n+2) or x=a(n+2), y=a(n+1), z=a(n), then x^2 -9*y*x +7*x*z +9*y^2 -9*z*y +z^2 = -45. - Alexander Samokrutov, Jul 02 2015

REFERENCES

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

FORMULA

a(n) = 8a(n-1) - 8a(n-2) + a(n-3).

From Paul Barry, Jan 20 2005: (Start)

G.f.: (1-4*x-2*x^2)/(1-8*x+8*x^2-x^3).

a(n) = Sum_{k=0..n} binomial(2n-k, 2k)*2^(2n-3k).

a(n) = 1 - 0^n + Sum_{k=0..2n} binomial(4n-k-1, k). (End)

a(n) = 1-(1/5)*sqrt(5)*{[(7/2)-(3/2)*sqrt(5))^n+[(7/2)+(3/2)*sqrt(5)]^n}, with n>=0. - Paolo P. Lava, Dec 01 2008

MAPLE

with(combinat): for n from 0 to 30 do printf(`%d, `, fibonacci(4*n)+1) od: # James A. Sellers, Mar 01 2003

MATHEMATICA

Fibonacci[4*Range[0, 30]]+1 (* or *) LinearRecurrence[{8, -8, 1}, {1, 4, 22}, 30] (* Harvey P. Dale, Feb 26 2015 *)

PROG

(MAGMA) [Fibonacci(4*n)+1: n in [0..30]]; // Vincenzo Librandi, Apr 20 2011

(PARI) Vec((1-4*x-2*x^2)/((1-x)*(1-7*x+x^2)) + O(x^30)) \\ Colin Barker, Dec 23 2014

(Sage) [fibonacci(4*n)+1 for n in (0..30)] # G. C. Greubel, Jul 15 2019

(GAP) List([0..30], n-> Fibonacci(4*n)+1); # G. C. Greubel, Jul 15 2019

CROSSREFS

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

Sequence in context: A104991 A027391 A134988 * A222012 A057834 A196795

Adjacent sequences:  A080999 A081000 A081001 * A081003 A081004 A081005

KEYWORD

nonn,easy

AUTHOR

R. K. Guy, Mar 01 2003

EXTENSIONS

More terms from James A. Sellers, Mar 01 2003

STATUS

approved

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Last modified May 26 00:32 EDT 2020. Contains 334613 sequences. (Running on oeis4.)