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A060645 a(0) = 0, a(1) = 4, then a(n) = 18*a(n-1) - a(n-2). 10
0, 4, 72, 1292, 23184, 416020, 7465176, 133957148, 2403763488, 43133785636, 774004377960, 13888945017644, 249227005939632, 4472197161895732, 80250321908183544, 1440033597185408060, 25840354427429161536 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
LINKS
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
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
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
Cf. A023039.
Sequence in context: A358295 A100521 A111868 * A363987 A203073 A231033
KEYWORD
nonn,easy
AUTHOR
Lekraj Beedassy, Apr 17 2001
EXTENSIONS
More terms from James A. Sellers, Apr 19 2001
Entry revised by N. J. A. Sloane, Aug 13 2006
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)