login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A132584 a(0)=0, a(1)=4; for n > 1, a(n) = 18*a(n-1) - a(n-2) + 8. 5
0, 4, 80, 1444, 25920, 465124, 8346320, 149768644, 2687489280, 48225038404, 865363202000, 15528312597604, 278644263554880, 5000068431390244, 89722587501469520, 1610006506595061124, 28890394531209630720, 518417095055178291844, 9302617316461999622480 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The old definition given for this sequence was "Sequence allows us to find X values of the equation: X(X + 1) - 5*Y^2 = 0".

With this old definition, if X = a(n), then Y = A207832(n). Now, with u = 2X+1, this Diophantine equation becomes the Pell-Fermat equation u^2 - 20*Y^2 = 1, and then, u = A023039(n) and Y = A207832(n). - Bernard Schott, Jan 25 2023

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

FORMULA

a(n) = -1/2 + (1/4)*(9-4*sqrt(5))^n + (1/4)*(9+4*sqrt(5))^n. - Paolo P. Lava, Oct 07 2008

a(n) = (A023039(n) - 1)/2. - Max Alekseyev, Nov 13 2009

G.f.: -4*x*(x+1)/((x-1)*(x^2-18*x+1)). - Colin Barker, Oct 24 2012

From Amiram Eldar, Jan 11 2022: (Start)

a(n) = 5*Fibonacci(3*n)^2/4 - 1 if n is odd and 5*Fibonacci(3*n)^2/4 if n is even.

A000217(a(n)) = A292443(n). (End)

a(n) = (Lucas(6*n)-2)/4. - Jeffrey Shallit, Jan 20 2023

a(n) = 4 * A049683(n). - Alois P. Heinz, Jan 20 2023

MATHEMATICA

LinearRecurrence[{19, -19, 1}, {0, 4, 80}, 40] (* Vincenzo Librandi, Dec 24 2018 *)

PROG

(Magma) I:=[0, 4, 80]; [n le 3 select I[n] else 18*Self(n-1)-Self(n-2)+8: n in [1..30]]; // Vincenzo Librandi, Dec 24 2018

CROSSREFS

Cf. A000032, A000045, A000217, A007654, A023039, A049683, A292443, A207832.

Sequence in context: A054322 A114488 A055787 * A277074 A012127 A189791

Adjacent sequences: A132581 A132582 A132583 * A132585 A132586 A132587

KEYWORD

nonn,easy,changed

AUTHOR

Mohamed Bouhamida, Nov 14 2007

EXTENSIONS

More terms from Max Alekseyev, Nov 13 2009

New definition by Antti Karttunen, Oct 24 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 8 03:07 EST 2023. Contains 360134 sequences. (Running on oeis4.)