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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
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).
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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 08 2002
STATUS
approved