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A077413 Bisection (odd part) of Chebyshev sequence with Diophantine property. 6
2, 13, 76, 443, 2582, 15049, 87712, 511223, 2979626, 17366533, 101219572, 589950899, 3438485822, 20040964033, 116807298376, 680802826223, 3968009658962, 23127255127549, 134795521106332, 785645871510443, 4579079707956326, 26688832376227513, 155553914549408752 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 6 14:15 EST 2019. Contains 329806 sequences. (Running on oeis4.)