0,2

Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(6) - 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(6). - Benoit Cloitre, Feb 24, 2004

(sqrt(2)+sqrt(3))^(2*n)=a(n)+A001078(n)*sqrt(6). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 12 2008

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 374.

V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, Paris, 1952, p. 281.

T. D. Noe, Table of n, a(n) for n=0..200

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Tanya Khovanova, Recursive Sequences

L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.

L. Euler, De solutione problematum diophanteorum per numeros integros, par. 18

Index entries for sequences related to Chebyshev polynomials.

For all members x of the sequence, 6*x^2 -6 is a square. Lim. n-> Inf. a(n)/a(n-1) = 5 + 2*Sqrt(6). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002

a(n) = T(n, 5) = (S(n, 10)-S(n-2, 10))/2 with S(n, x) := U(n, x/2), and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 10)= A004189(n+1).

a(n) = sqrt(1+24*A004189(n)^2) (cf. Richardson comment).

a(n)a(n+3) - a(n+1)a(n+2) = 240. - R. Stephan, Jun 06 2005

Chebyshev's polynomials T(n,x) evaluated at x=5.

G.f.: (1-5*x)/(1-10*x+x^2). a(n)= ((5+2*sqrt(6))^n + (5-2*sqrt(6))^n)/2.

a(-n)=a(n).

a(n+1)=5*a(n)+2*(6*a(n)^2-6)^0.5 - Richard Choulet (richardchoulet(AT)yahoo.fr), Sep 19 2007

a(n+1) = 2*A054320(n) + 3*A138288(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 12 2008

a(n) = Cosh[2n*ArcSinh[Sqrt[2]]] - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008

A001079:=-(-1+5*z)/(1-10*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

(PARI) {a(n)=subst(poltchebi(n), 'x, 5)} /* Michael Somos Sep 05 2006 */

(PARI) {a(n)=real((5+2*quadgen(24))^n)} /* Michael Somos Sep 05 2006 */

(PARI) {a(n)=n=abs(n); polsym(1-10*x+x^2, n)[n+1]/2} /* Michael Somos Sep 05 2006 */

Cf. A004189.

Cf. A001078 A046173 A046172 A036353.

Cf. A138281.

Sequence in context: A083224 A093188 A096596 this_sequence A081474 A112241 A116873

Adjacent sequences: A001076 A001077 A001078 this_sequence A001080 A001081 A001082

nonn

njas

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