A077234 Bisection (odd part) of Chebyshev sequence with Diophantine property.

2, 9, 34, 127, 474, 1769, 6602, 24639, 91954, 343177, 1280754, 4779839, 17838602, 66574569, 248459674, 927264127, 3460596834, 12915123209, 48199896002, 179884460799, 671337947194, 2505467327977, 9350531364714, 34896658130879, 130236101158802, 486047746504329

Offset

0,1

Comments

-3*a(n)^2 + b(n)^2 = 13, with the companion sequence b(n) = A077235(n).

The even part is A054491(n) with Diophantine companion A077236(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 (4,-1).

Formula

a(n) = 2*S(n, 4)+S(n-1, 4), with S(n, x) = U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(-1, x) = 0 and S(n, 4) = A001353(n+1).

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

a(n) = 4*a(n-1)-a(n-2) with a(0)=2 and a(1)=9. - Philippe Deléham, Nov 16 2008

E.g.f.: exp(2*x)*(6*cosh(sqrt(3)*x) + 5*sqrt(3)*sinh(sqrt(3)*x))/3. - Stefano Spezia, Oct 19 2023

Example

3*a(1)^2 + 13 = 3*81+13 = 256 = 16^2 = A077235(1)^2.

Prog

(PARI) Vec((2+x)/(1-4*x+x^2) + O(x^50)) \\ Colin Barker, Jun 16 2015

Crossrefs

Cf. A001353, A049310, A054491, A077235, A077236, A077237 (even and odd parts).

Sequence in context: A120989 A280309 A010763 * A091526 A274750 A204444

Adjacent sequences: A077231 A077232 A077233 * A077235 A077236 A077237

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Status

approved