1,2

Bisection (even part) of Chebyshev sequence with Diophantine property.

(3*b(n))^2 - 2*(2*a(n+1))^2 = 1 with companion sequence b(n)=A077420(n),n>=0.

Sequence also refers to inradius of primitive Pythagorean triangles with consecutive legs, odd followed by even. - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 23 2003

Tanya Khovanova, Recursive Sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Index entries for sequences related to Chebyshev polynomials.

a(n)=34*a(n-1)-a(n-2), a(0)=-1, a(1)=1.

a(n+1)= S(2*n, 6)= S(n, 34) + S(n-1, 34), n>=1, with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(n, 34)=A029547(n).

G.f.: x*(1+x)/(1-34*x+x^2).

a(n+1)= sum((-1)^k*binomial(2*n-k, k)*6^(2*(n-k)), k=0..n), n>=0.

a(n)=A001109(2n+1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 23 2003

Define f[x,s] = s x + Sqrt[(s^2-1)x^2+1]; f[0,s]=0. a(n) = f[f[a(n-1),3],3]. - Marcos Carreira, Dec 27 2006

(1) a(n) = (sqrt(2)/8)(3+2*sqrt(2))*(17+12*sqrt(2))^(n-1) -(sqrt(2)/8)(3-2*sqrt(2))*(17-12*sqrt(2))^(n-1) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 22 2008

(2) a(n) = (sqrt(2)/8)*(17+12*sqrt(2))^(n-1/2) -(sqrt(2)/8)*(17-12*sqrt(2))^(n-1/2) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 22 2008

(3) a(n) = (sqrt(2)/8)*(3+2*sqrt(2))^(2n-1) -(sqrt(2)/8)*(3-2*sqrt(2))^(2n-1) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 22 2008

(4) a(n) = (sqrt(2)/8)*(1+sqrt(2))^(4n-2) -(sqrt(2)/8)*(1-sqrt(2))^(4n-2) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 22 2008

(5) a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 22 2008

a(n+1)=17*a(n)+6*(8*a(n)^2+1)^0.5 for n>=0 [From Richard Choulet (richardchoulet(AT)yahoo.fr), May 01 2009]

q=9; s=0; lst={}; Do[s+=n; If[Sqrt[q*s+1]==Floor[Sqrt[q*s+1]], AppendTo[lst, Sqrt[q*s+1]]], {n, 0, 9!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 02 2009]

Cf. A008844, A046177.

Cf. A001109.

Sequence in context: A002453 A049395 A115492 this_sequence A029546 A126158 A095153

Adjacent sequences: A046173 A046174 A046175 this_sequence A046177 A046178 A046179

nonn

Eric Weisstein (eric(AT)weisstein.com)

Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002