1,2

Chebyshev S-sequence with diophantine property.

4*b(n)^2 - 3*a(n)^2 = 1 with b(n)=A001570(n), n>=0.

y satisfying the Pellian x^2 - 3*y^2=1, for even x given by A094347(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 03 2004

a(n) = L(n,-14)*(-1)^n, where L is defined as in A108299; see also A001570 for L(n,+14). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005

Product x*y, where the pair (x, y) solves for x^2 - 3y^2 = -2, i.e., a(n)=A001834(n)*A001835(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 13 2006

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.

T. N. E. Greville, Table for third-degree spline interpolations with equally spaced arguments, Math. Comp., 24 (1970), 179-.

W. D. Hoskins, Table for third-degree spline interpolation using equi-spaced knots, Math. Comp., 25 (1971), 797-801.

J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.

F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.

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)=2*A001921(n)+1.

a(n) = 14*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.

a(n) = S(n, 14) + S(n-1, 14) = S(2*n, 4) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(n, 14)=A007655(n+1) and S(n, 4)=A001353(n+1).

G.f.: x*(1+x)/(1-14*x+x^2).

a(n) = (ap^(2*n+1) - am^(2*n+1))/(ap - am) with ap := 2+sqrt(3) and am := 2-sqrt(3).

a(n+1)= sum(((-1)^k)*binomial(2*n-k, k)*16^(n-k), k=0..n), n>=0.

a(n) = sqrt((4*A001570(n-1)^2 - 1)/3).

a(n) ~ 1/6*sqrt(3)*(2 + sqrt(3))^(2*n-1) - Joe Keane (jgk(AT)jgk.org), May 15 2002

4*a(n+1) = (A001834(n))^2 + 4*(A001835(n+1))^2 - (A001835(n))^2. E.g. 4*a(3) = 4*209 = 19^2 + 4*11^2 - 3^2 = (A001834(2))^2 + 4*(A001835(3))^2 - A001835(2))^2. Generating floretion: 'i + 2'j + 3'k + i' + 2j' + 3k' + 4'ii' + 3'jj' + 4'kk' + 3'ij' + 3'ji' + 'jk' + 'kj' + 4e - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Dec 04 2004

Define f[x,s] = s x + Sqrt[(s^2-1)x^2+1]; f[0,s]=0. a(n) = f[a(n-1),7] + f[a(n-2),7]. - Marcos Carreira, Dec 27 2006

Cf. A036428, A046184.

Cf. A077416 with companion A077417.

Sequence in context: A078265 A089138 A051813 this_sequence A122572 A067560 A019553

Adjacent sequences: A028227 A028228 A028229 this_sequence A028231 A028232 A028233

nonn

njas

More terms from Eric Weisstein (eric(AT)weisstein.com)

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