A077246 Bisection (even part) of Chebyshev sequence with Diophantine property.

2, 13, 102, 803, 6322, 49773, 391862, 3085123, 24289122, 191227853, 1505533702, 11853041763, 93318800402, 734697361453, 5784260091222, 45539383368323, 358530806855362, 2822707071474573, 22223125764941222

Offset: 0

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

Comments

3*a(n)^2 - 5*b(n)^2 = 7, with the companion sequence b(n)= A077245(n).

The odd part is A077244(n) with Diophantine companion A077243(n).

Examples

			13 = a(1) = sqrt((5*A077245(1)^2 + 7)/3) = sqrt((5*10^2 + 7)/3) = sqrt(169) = 13.
		

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (8,-1).

Programs

Mathematica

LinearRecurrence[{8,-1},{2,13},30] (* Harvey P. Dale, Apr 30 2012 *)

Formula

a(n)= 8*a(n-1) - a(n-2), a(-1) := 3, a(0)=2.

a(n)= (T(n+1, 4)+2*T(n, 4))/3, with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 4)= A001091(n).

G.f.: (2-3*x)/(1-8*x+x^2).