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A077234 Bisection (odd part) of Chebyshev sequence with Diophantine property. 5
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 (list; graph; refs; listen; history; text; internal format)
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

a(n) = -(5/6)*sqrt(3)*[2-sqrt(3)]^n+(5/6)*sqrt(3)*[2+sqrt(3)]^n+[2-sqrt(3)]^n+[2+sqrt(3)]^n, with n>=0. - Paolo P. Lava, Nov 20 2008

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. 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

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Last modified March 25 05:37 EDT 2017. Contains 284036 sequences.