A077235 Bisection (odd part) of Chebyshev sequence with Diophantine property.

5, 16, 59, 220, 821, 3064, 11435, 42676, 159269, 594400, 2218331, 8278924, 30897365, 115310536, 430344779, 1606068580, 5993929541, 22369649584, 83484668795, 311569025596, 1162791433589, 4339596708760, 16195595401451, 60442784897044, 225575544186725

Offset: 0

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

Comments

a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n) = A077234(n).

The even part is A077236(n) with Diophantine companion A054491(n).

Examples

			16 = a(1) = sqrt(3*A077234(1)^2 + 13) = sqrt(3*9^2 + 13)= sqrt(256) = 16.
		

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 (4,-1)

Crossrefs

Cf. A077238 (even and odd parts).

Programs

PARI

Vec((5-4*x)/(1-4*x+x^2) + O(x^100)) \\ Colin Barker, Jun 16 2015

Formula

a(n) = 2*T(n+1, 2)+T(n, 2), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 2)= A001075(n).

G.f.: (5-4*x)/(1-4*x+x^2).

a(n) = 4*a(n-1)-a(n-2) with a(0)=5 and a(1)=16. - Philippe Deléham, Nov 16 2008