0,3

a(2*n+1) with b(2*n+1) := A001077(2*n+1), n>=0, give all (positive integer) solutions to Pell equation b^2 - 5*a^2 = -1, a(2*n) with b(2*n) := A001077(2*n), n>=1, give all (positive integer) solutions to Pell equation b^2 - 5*a^2 = +1 (cf. Emerson reference).

Bisection: a(2*n+1)= T(2*n+1,sqrt(5))/sqrt(5)= A007805(n), n>=0, and a(2*n)=4*S(n-1,18),n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first,resp. second kind. S(-1,x)=0. See A053120, resp. A049310. S(n,18)=A049660(n+1).

Apart from initial terms, this is the Pisot sequence E(4,17), a(n)=[ a(n-1)^2/a(n-2)+1/2 ].

This is also the Horadam sequence (0,1,1,4), having the recurrence relation a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 4, r = 1. a(n) / a(n-1) converges to 5^1/2 + 2 as n approaches infinity. 5^1/2 + 2 can also be written as (2 * Phi) + 1 and Phi^2 + Phi. - Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003

Numerators in continued fraction [2, 4, 4, 4,...] = (1, 4, 17, 72,...) = numerators of continued fraction [4, 4, 4,...]; where the convergents to [4, 4, 4,...] = (1/4, 4/17, 17/72,...). Let X = the 2 X 2 matrix [0, 1; 1, 4]; then X^n = [a(n-1), a(n); a(n), a(n+1)]; e.g. X^3 = [4, 17; 17, 72]. Let C = the limit of a(n)/a(n-1) = 2 + sqrt(5) = 4.236067977...; then C^n = a(n+1) + (1/C)*a(n), where (1/C) = .236067977.... Example: C^3 = 76.01315556..., = 72 + 17*(.2360679....). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 15 2007

Sqrt(5) = 4/2 + 4/17 + 4/(17*305) + 4/(305*5473) + 4/(5473*98209) +... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 15 2007

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.

D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

E. I. Emerson, Recurrent sequences in the equation DQ^2=R^2+N, Fib. Quart., 7 (1969), 231-242, Thm. 1, p. 233.

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

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 23.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 398

Index entries for sequences related to Chebyshev polynomials.

Zerinvary Lajos, Sage Notebooks

a(n) = 4a(n-1) + a(n-2), n>1. a(0)=0, a(1)=1. G.f.: x/(1-4*x-x^2).

a(n)=((2+sqrt(5))^n - (2-sqrt(5))^n)/(2*sqrt(5)).

a(n) = ((-i)^(n-1))*S(n-1, 4*i), with i^2 =-1, and S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0.

a(n)=F(3n)/F(3), with F(n) Fibonacci numbers. - Mario Catalani (mario.catalani(AT)unito.it), Jul 24 2003

a(n)=sum{i=0..n, sum{j=0..n, Fib(i+j)*n!/(i!j!(n-i-j)!)/2}} - Paul Barry (pbarry(AT)wit.ie), Feb 06 2004

E.g.f.: exp(2*x)*sinh(sqrt(5)*x)/sqrt(5). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 01 2004

a(n) = F(1) + F(4) + F(7) + ... + F(3n-2), for n>0.

Conjecture: 2a(n+1) = a(n+2) - A001077(n+1); Sequences (a(n)), A001077 generated by floretion: 'ii' + 'jj' - 'kk' + 0.5'ik' + 0.5'ki' - e - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 28 2004

a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j, k)F(j)/2}} - Paul Barry (pbarry(AT)wit.ie), Feb 14 2005

a(n) = A048876(n) - A048875(n) - Creighton Dement (crowdog(AT)t-online.de), Mar 19 2005

Let M = {{0, 1}, {1, 4}}, v[1] = {0, 1}, v[n] = M.v[n - 1]; then a(n) = v[n][[1]]. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 29 2005 - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006

a(n)=F(n, 4), the n-th Fibonacci polynomial evaluated at x=4. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006

[A015448(n), a(n)] = [1,4; 1,3]^n * [1,0] - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2008

1 2 9 38 161 (A001077)

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

0 1 4 17 72 (A001076)

K:=1/(1+4*z-z^2): Kser:=series(K, z=0, 30): seq(abs(coeff(Kser, z, n)), n= -1..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 08 2007

A001076:=-1/(-1+4*z+z**2); [Conjectured by S. Plouffe in his 1992 dissertation.]

with(combinat): a:=n->fibonacci(n, 4)-4*fibonacci(n-1, 4): seq(a(n), n=2..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008

(Mupad) numlib::fibonacci(3*n)/2 $ n = 0..30; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008

sage: from sage.combinat.sloane_functions import recur_gen3 sage: it = recur_gen3(0, 1, 4, 4, 1, 0) sage: [it.next() for i in xrange(1, 32)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008

Cf. A001077, A015448, A033887.

A001076(n)=F(3n)/2, where F=A000045 (the Fibonacci sequence).

Cf. A049660, A007805.

Partial sums of A033887. First differences of A049652. Bisection of A059973.

Third column of array A028412.

Cf. A015448.

Sequence in context: A136792 A108929 A022031 this_sequence A122451 A113442 A085732

Adjacent sequences: A001073 A001074 A001075 this_sequence A001077 A001078 A001079

nonn,easy,cofr,nice

njas

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