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

0,1

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).

Links

Table of n, a(n) for n=0..18.

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (8,-1).

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).

Example

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

Mathematica

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

Crossrefs

Sequence in context: A330349 A234299 A301355 * A365148 A371578 A266906

Adjacent sequences: A077243 A077244 A077245 * A077247 A077248 A077249

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Status

approved