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

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

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).

Examples

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

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).

Programs

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 *)

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

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).