A060645 a(0) = 0, a(1) = 4, then a(n) = 18*a(n-1) - a(n-2).

0, 4, 72, 1292, 23184, 416020, 7465176, 133957148, 2403763488, 43133785636, 774004377960, 13888945017644, 249227005939632, 4472197161895732, 80250321908183544, 1440033597185408060, 25840354427429161536

Offset: 0

Author

Lekraj Beedassy, Apr 17 2001

Keywords

• nonn
• easy

Comments

This sequence gives the values of y in solutions of the Diophantine equation x^2 - 5*y^2 = 1, the third simplest case of the Pell-Fermat type. The corresponding x values are in A023039.

Numbers k such that 5*k^2 = floor(sqrt(5)*k*ceiling(sqrt(5)*k)). - Benoit Cloitre, May 10 2003

Examples

			Given a(1) = 4, a(2) = 72 we have, for instance, a(4) = 18*a(3) - a(2) = 18*{18*a(2) - a(1)} - a(2), i.e., a(4) = 323*a(2) - 18*a(1) = 323*72 - 18*4 = 23184.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.
• Tanya Khovanova, Recursive Sequences
• John Robertson, Home page
• Index entries for linear recurrences with constant coefficients, signature (18,-1).

Crossrefs

Cf. A023039.

Programs

Maple

A060645 := proc(n) option remember: if n=1 then RETURN(4) fi: if n=2 then RETURN(72) fi: 18*A060645(n -1)- A060645(n-2): end: for n from 1 to 30 do printf(`%d,`, A060645(n)) od:

Mathematica

CoefficientList[ Series[4x/(1 - 18x + x^2), {x, 0, 16}], x] (* Robert G. Wilson v *)
LinearRecurrence[{18, -1} {0, 4}, 50] (* Sture Sjöstedt, Nov 29 2011 *)
Table[4 ChebyshevU[-1 + n, 9], {n, 0, 16}] (* Herbert Kociemba, Jun 05 2022 *)

PARI

g(n,k) = for(y=0,n,x=k*y^2+1;if(issquare(x),print1(y",")))

PARI

a(n)=fibonacci(6*n)/2 \\ Benoit Cloitre

PARI

for (i=1,10000,if(Mod(sigma(5*i^2+1),2)==1,print1(i,",")))

PARI

{ for (n=0, 200, write("b060645.txt", n, " ", fibonacci(6*n)/2); ) } \\ Harry J. Smith, Jul 09 2009

Formula

a(n) = 18*a(n-1) - a(n-2), with a(1) = denominator of continued fraction [2; 4] and a(2) = denominator of [2; 4, 4, 4].

G.f.: 4x/(1-18*x+x^2). - Cino Hilliard, Feb 02 2006

a(n) may be computed either as (i) the denominator of the (2n-1)-th convergent of the continued fraction [2; 4, 4, 4, ...] = sqrt(5), or (ii) as the coefficient of sqrt(5) in (9+sqrt(5))^n.

Numbers k such that sigma(5*k^2 + 1) mod 2 = 1. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004

a(n) = 4*A049660(n), a(n) = A000045(6*n)/2. - Benoit Cloitre, Feb 03 2006

a(n) = 17*(a(n-1) + a(n-2)) - a(n-3) = 19*(a(n-1) - a(n-2)) + a(n-3). - Mohamed Bouhamida, Sep 20 2006

From Johannes W. Meijer, Jul 01 2010: (Start)

Limit_{k->infinity} a(n+k)/a(k) = A023039(n) + A060645(n)*sqrt(5).

Limit_{n->infinity} A023039(n)/a(n) = sqrt(5). (End)

a(n) = Fibonacci(6*n)/2. - Gary Detlefs, Apr 02 2012

a(n) = 4*S(n-1, 18), with Chebyshev's S-polynomials. See A049310. S(-1, x)= 0. - Wolfdieter Lang, Aug 24 2014

Extensions

More terms from James Sellers, Apr 19 2001

Entry revised by N. J. A. Sloane, Aug 13 2006