The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A081002 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A081002

a(n) = Fibonacci(4n) + 1, or Fibonacci(2n-1)*Lucas(2n+1).
(history; published version)

#40 by Joerg Arndt at Mon Jan 01 11:07:23 EST 2024

STATUS

editing

approved

#39 by Paolo P. Lava at Mon Jan 01 11:04:19 EST 2024

FORMULA

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

STATUS

approved

editing

#38 by Charles R Greathouse IV at Thu Sep 08 08:45:09 EDT 2022

PROG

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

Discussion

Thu Sep 08

08:45

OEIS Server: https://oeis.org/edit/global/2944

#37 by Bruno Berselli at Mon Jul 15 02:58:48 EDT 2019

STATUS

reviewed

approved

#36 by Michel Marcus at Mon Jul 15 02:56:31 EDT 2019

STATUS

proposed

reviewed

#35 by G. C. Greubel at Mon Jul 15 02:55:15 EDT 2019

STATUS

editing

proposed

#34 by G. C. Greubel at Mon Jul 15 02:55:09 EDT 2019

NAME

a(n) = Fibonacci(4n) + 1, or Fibonacci(2n-1)*Lucas(2n+1).

FORMULA

G.f.: (1-4x-2x^2)/(1-8x+8x^2-x^3); a(n) = sum{k=0..n, binomial(2n-k, 2k)2^(2n-3k)}; a(n) = sum{k=0..2n, binomial(4n-k-1, k)}+1-0^n. - Paul Barry, Jan 20 2005

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)

MAPLE

with(combinat): for n from 0 to 25 30 do printf(`%d, `, fibonacci(4*n)+1) od: # James A. Sellers, Mar 01 2003

MATHEMATICA

Fibonacci[4*Range[0, 2030]]+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..5030]]; // Vincenzo Librandi, Apr 20 2011

(PARI) Vec((21-4*x^-2+4*x-^2)/((1)/((-x-)*(1)*(x^2-7*x+1x^2)) + O(x^10030)) \\ 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

STATUS

approved

editing

#33 by Giovanni Resta at Tue Jul 14 17:04:29 EDT 2015

STATUS

editing

approved

#32 by Giovanni Resta at Tue Jul 14 17:02:21 EDT 2015

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

Discussion

Tue Jul 14

17:03

Giovanni Resta: double edit, (my error.)

#31 by Giovanni Resta at Tue Jul 14 17:00:18 EDT 2015

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

STATUS

proposed

editing