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A023039 a(n) = 18a(n-1) - a(n-2). 25
1, 9, 161, 2889, 51841, 930249, 16692641, 299537289, 5374978561, 96450076809, 1730726404001, 31056625195209, 557288527109761, 10000136862780489, 179445175002939041, 3220013013190122249, 57780789062419261441 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The primitive Heronian triangle 3*a(n) +/- 2, 4*a(n) has the latter side cut into 2*a(n) +/- 3 by the corresponding altitude and has area 10*a(n)*A060645(n). - Lekraj Beedassy, Jun 25 2002

Chebyshev polynomials T(n,x) evaluated at x=9.

The a(n) give all (unsigned, integer) solutions of Pell equation a(n)^2 - 80*b(n)^2 = +1 with b(n)=A049660(n), n>=0.

Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(5). - Benoit Cloitre, Feb 14 2004

Appears to give all solutions >1 to the equation: x^2=ceiling(x*r*floor(x/r)) where r=sqrt(5). - Benoit Cloitre, Feb 24 2004

For all members x of the sequence, 5*x^2 - 5 is a square, A004292(n)^2.

The a(n) are the x-values in the nonnegative integer solutions of x^2-5y^2=1, see A060645(n) for the corresponding y-values. - Sture Sjöstedt, Nov 29 2011

Rightmost digits alternate repeatedly: 1 and 9 in fact, a(2)= 18*9-1=1 (mod 10); a(3)=18*1-9=9 (mod 10) therefore a(2n)=1 (mod 10); a(2n+1)=9 (mod 10). - Carmine Suriano, Oct 03 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (18,-1).

FORMULA

a(n) ~ 1/2*(sqrt(5) + 2)^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002

Lim. n-> Inf. a(n)/a(n-1) = phi^6 = 9 + 4*Sqrt(5). - Gregory V. Richardson, Oct 13 2002

a(n) = T(n, 9) = (S(n, 18)-S(n-2, 18))/2, with S(n, x) := U(n, x/2) and T(n, x), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 18)=A049660(n+1).

a(n) = sqrt(80*A049660(n)^2 + 1) (cf. Richardson comment).

a(n) = ((9+4*sqrt(5))^n + (9-4*sqrt(5))^n)/2.

G.f.: (1-9*x)/(1-18*x+x^2).

a(n) = cosh[2n*arcsinh[2]]. - Herbert Kociemba, Apr 24 2008

a(n) = A001077(2*n). - Michael Somos, Aug 11 2009

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

a(n) = 2*A167808(6*n+1) - A167808(6*n+3).

Limit(a(n+k)/a(k), k=infinity) = a(n) + A060645(n)*sqrt(5).

Limit(a(n)/A060645(n), n=infinity) = sqrt(5).

(End)

a(n) = 1/2*Lucas(6*n) = 1/2*(sqrt(5) + 2)^(2*n) + 1/2*(sqrt(5) - 2)^(2*n).

Sum {n >= 1} 1/( a(n) - 5/a(n) ) = 1/8. Compare with A005248, A002878 and A075796. - Peter Bala, Nov 29 2013

a(n) = 2*A115032(n-1) - 1 =  S(n, 18) - 9*S(n-1, 18), with A115032(-1) = 1, and see the above formula with S(n, 18) using its recurrence. - Wolfdieter Lang, Aug 22 2014

EXAMPLE

G.f. = 1 + 9*x + 161*x^2 + 2889*x^3 + 51841*x4 + 930249*x^5 + 16692641*x^6 + ...

MATHEMATICA

LinearRecurrence[{18, -1}, {1, 9}, 50] (* Sture Sjöstedt, Nov 29 2011 *)

PROG

(PARI) {a(n) = fibonacci(6*n) / 2 + fibonacci(6*n - 1)}; /* Michael Somos, Aug 11 2009 */

(MAGMA) I:=[1, 9]; [n le 2 select I[n] else 18*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 13 2012

CROSSREFS

Bisection of A001077.

Row 2 of array A188645.

Cf. A115032. - Wolfdieter Lang, Aug 22 2014

Sequence in context: A217392 A062232 A020523 * A243682 A159831 A133793

Adjacent sequences:  A023036 A023037 A023038 * A023040 A023041 A023042

KEYWORD

nonn,easy

AUTHOR

David W. Wilson

EXTENSIONS

More terms from Joe Keane (jgk(AT)jgk.org), May 15 2002

Chebyshev and Pell comments from Wolfdieter Lang, Nov 08 2002

Sture Sjöstedt comment corrected and reformulated. - Wolfdieter Lang, Aug 24 2014

STATUS

approved

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Last modified May 23 07:01 EDT 2017. Contains 286909 sequences.