0,2

a(2*n+1) with b(2*n+1) := A001076(2*n+1), n>=0, give all (positive integer) solutions to Pell equation a^2 - 5*b^2 = -1.

a(2*n) with b(2*n) := A001076(2*n), n>=1, give all (positive integer) solutions to Pell equation a^2 - 5*b^2 = +1 (see Emerson reference).

Bisection: a(2*n)= T(n,9)= A023039(n), n>=0, and a(2*n+1)=2*S(2*n,2*sqrt(5)),n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. See A053120, resp. A049310.

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.

E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Ex.1, p. 237-8.

V. Th\'{e}bault, Les R\'{e}cr\'{e}ations Math\'{e}matiques. Gauthier-Villars, Paris, 1952, p. 282.

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

Index entries for sequences related to Chebyshev polynomials.

G.f.: (1-2*x)/(1-4*x-x^2); a(n)=4*a(n-1)+a(n-2), a(0)=1, a(1)=2; a(n)=[ (2+sqrt(5))^n + (2-sqrt(5))^n ]/2.

Lim. n-> Inf. a(n)/a(n-1) = phi^3 = 2 + Sqrt(5). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002

a(n) = ((-i)^n)*T(n, 2*i), with T(n, x) Chebyshev's polynomials of the first kind A053120, and i^2 = -1.

Binomial transform of A084057. - Paul Barry (pbarry(AT)wit.ie), May 10 2003

E.g.f.: exp(2x)cosh(sqrt(5)x) - Paul Barry (pbarry(AT)wit.ie), May 10 2003

a(n)=sum{k=0..floor(n/2), C(n, 2k)5^k2^(n-2k)} - Paul Barry (pbarry(AT)wit.ie), Nov 15 2003

a(n) = 4*a(n-1) + a(n-2) when n > 2; a[1] = 1, a[2] = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 25 2004

a(n) = A001076(n+1) - 2*A001076(n) = A097924(n) - A015448(n+1); a(n+1) = A097924(n) + 2*A001076(n) = A097924(n) + 2(A048876(n) - A048875(n)) - Creighton Dement (crowdog(AT)t-online.de), Mar 19 2005

a(n) = F(3*n)/2 + F(3*n-1) where F() = Fibonacci numbers A000045. - Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Apr 28 2007

1 2 9 38 161 (A001077)

-,-,-,--,---, ...

0 1 4 17 72 (A001076)

A001077:=(-1+2*z)/(-1+4*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

with(combinat): a:=n->fibonacci(n, 4)-2*fibonacci(n-1, 4): seq(a(n), n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008

A001077(n)=A014448(n)/2.

Cf. A001076.

Cf. A023039, A049629.

Sequence in context: A007224 A037489 A037569 this_sequence A105484 A057647 A069724

Adjacent sequences: A001074 A001075 A001076 this_sequence A001078 A001079 A001080

nonn,easy,cofr,nice

njas

Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jan 10 2003