A077413 Bisection (odd part) of Chebyshev sequence with Diophantine property.

2, 13, 76, 443, 2582, 15049, 87712, 511223, 2979626, 17366533, 101219572, 589950899, 3438485822, 20040964033, 116807298376, 680802826223, 3968009658962, 23127255127549, 134795521106332, 785645871510443, 4579079707956326, 26688832376227513, 155553914549408752

Offset: 0

Author

Wolfdieter Lang, Nov 08 2002

Keywords

• nonn
• easy

Comments

-8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n) = A077239(n).

The even part is A054488(n) with Diophantine companion A077240(n).

Examples

			8*a(1)^2 + 17 = 8*13^2+17 = 1369 = 37^2 = A077239(1)^2.
G.f. = 2 + 13*x + 76*x^2 + 443*x^3 + 2582*x^4 + ... - _Michael Somos_, Jul 30 2024
		

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 (6,-1).

Crossrefs

Cf. A077241 (even and odd parts), A001109, A054488.

Programs

Magma

I:=[2,13]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018

Mathematica

LinearRecurrence[{6,-1}, {2,13}, 30] (* or *) CoefficientList[Series[ (2+x)/(1-6*x+x^2), {x, 0, 50}], x] (* G. C. Greubel, Jan 18 2018 *)
a[ n_] := 2*ChebyshevU[n, 3] + ChebyshevU[n-1, 3]; (* Michael Somos, Jul 30 2024 *)

PARI

Vec((2+x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Jun 16 2015

PARI

{a(n) = 2*polchebyshev(n, 2, 3) + polchebyshev(n-1, 2, 3)}; /* Michael Somos, Jul 30 2024 */

PARI

{a(n) = my(w = 3 + quadgen(32)); imag(w^n + 2*w^(n+1))}; /* Michael Somos, Jul 30 2024 */

Formula

a(n) = 6*a(n-1) - a(n-2), a(-1)=-1, a(0)=2.

a(n) = 2*S(n, 6)+S(n-1, 6), with S(n, x) = U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 6) = A001109(n+1).

G.f.: (2+x)/(1-6*x+x^2).

a(n) = (((3-2*sqrt(2))^n*(-7+4*sqrt(2))+(3+2*sqrt(2))^n*(7+4*sqrt(2))))/(4*sqrt(2)). - Colin Barker, Oct 12 2015

a(n) = -A054488(-1-n) for all n in Z. - Michael Somos, Jul 30 2024