0,2

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 (blekraj(AT)yahoo.com), Jun 25 2002

Chebyshev's 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 (benoit7848c(AT)orange.fr), 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

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

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

For all members x of the sequence, 5*x^2 - 5 is a square. Lim. n-> Inf. a(n)/a(n-1) = phi^6 = 9 + 4*Sqrt(5). - Gregory V. Richardson (omomom(AT)hotmail.com), 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 (kociemba(AT)t-online.de), Apr 24 2008

1 + 9*x + 161*x^2 + 2889*x^3 + 51841*x^4 + 930249*x^5 + 16692641*x^6 + ... - Michael Somos Aug 11 2009

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

Bisection of A001077.

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

Sequence in context: A060348 A062232 A020523 this_sequence A159831 A133793 A084874

Adjacent sequences: A023036 A023037 A023038 this_sequence A023040 A023041 A023042

nonn

David W. Wilson (davidwwilson(AT)comcast.net)

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

Chebyshev and Pell comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002