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

0,1

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).

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).

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

Example

8*a(1)^2 + 17 = 8*13^2+17 = 1369 = 37^2 = A077239(1)^2.

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 *)

Prog

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

Crossrefs

Cf. A077241 (even and odd parts).

Sequence in context: A161130 A192700 A007509 * A024199 A037523 A037732

Adjacent sequences: A077410 A077411 A077412 * A077414 A077415 A077416

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Status

approved