A077251 Bisection (even part) of Chebyshev sequence with Diophantine property.

1, 12, 119, 1178, 11661, 115432, 1142659, 11311158, 111968921, 1108378052, 10971811599, 108609737938, 1075125567781, 10642645939872, 105351333830939, 1042870692369518, 10323355589864241, 102190685206272892, 1011583496472864679, 10013644279522373898

Offset

0,2

Comments

b(n)^2 - 24*a(n)^2 = 25, with the companion sequence b(n) = A077409(n).

The odd part is A077249(n) with Diophantine companion A077250(n).

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (10,-1).

Formula

a(n) = 10*a(n-1)- a(n-2), a(-1)=-2, a(0)=1.

a(n) = S(n, 10)+2*S(n-1, 10), with S(n, x) = U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310. S(n, 10)= A004189(n+1).

a(n) = sqrt((A077409(n)^2 - 25)/24).

G.f.: (1+2*x)/(1-10*x+x^2).

Example

24*a(1)^2 + 25 = 24*12^2 + 25 = 3481 = 59^2 = A077409(1)^2.

Mathematica

CoefficientList[Series[(2 z + 1)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

Prog

(PARI) Vec((1+2*x)/(1-10*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 11 2011
(PARI) a(n)=([0, 1; -1, 10]^n*[1; 12])[1, 1] \\ Charles R Greathouse IV, Jun 15 2015

Crossrefs

Sequence in context: A025132 A001712 A285232 * A289542 A075622 A153054

Adjacent sequences: A077248 A077249 A077250 * A077252 A077253 A077254

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Status

approved