A077249 Bisection (odd part) of Chebyshev sequence with Diophantine property.

2, 21, 208, 2059, 20382, 201761, 1997228, 19770519, 195707962, 1937309101, 19177383048, 189836521379, 1879187830742, 18602041786041, 184141230029668, 1822810258510639, 18043961355076722, 178616803292256581, 1768124071567489088, 17502623912382634299

Offset

0,1

Comments

-24*a(n)^2 + b(n)^2 = 25, with the companion sequence b(n) = A077250(n).

The even part is A077251(n) with Diophantine companion A077409(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) := -1, a(0)=2.

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

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

Example

24*a(1)^2 + 25 = 24*21^2+25 = 10609 = 103^2 = A077250(1)^2.

Mathematica

CoefficientList[Series[(z + 2)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10, -1}, {2, 21}, 40] (* Harvey P. Dale, Apr 08 2012 *)

Prog

(PARI) a(n)=if(n<0, 0, subst(-7*poltchebi(n)+11*poltchebi(n+1), x, 5)/24)
(PARI) a(n)=2*polchebyshev(n, 2, 5)+polchebyshev(n-1, 2, 5) \\ Charles R Greathouse IV, Jun 11 2011
(PARI) Vec((2+x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015

Crossrefs

Sequence in context: A365061 A110253 A185634 * A068070 A085953 A225614

Adjacent sequences: A077246 A077247 A077248 * A077250 A077251 A077252

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Status

approved