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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Index entries for sequences related to linear recurrences with constant coefficients

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)

1 + 2*x + 9*x^2 + 38*x^3 + 161*x^4 + 682*x^5 + 2889*x^6 + 12238*x^7 + ... - Michael Somos Aug 11 2009

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

(Other) sage: [lucas_number2(n, 4, -1)/2 for n in xrange(0, 23)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009]

(PARI) {a(n) = fibonacci(3*n) / 2 + fibonacci(3*n - 1)} - Michael Somos Aug 11 2009

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

Cf. A001076.

Cf. A023039, A049629.

Sequence in context: A007224 A037489 A037569 this_sequence A150993 A150994 A150995

Adjacent sequences: A001074 A001075 A001076 this_sequence A001078 A001079 A001080

nonn,easy,cofr,nice

N. J. A. Sloane (njas(AT)research.att.com).

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