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

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

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).

Examples

			24*a(1)^2 + 25 = 24*12^2 + 25 = 3481 = 59^2 = A077409(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[(2 z + 1)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

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

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).