OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..800
Wolfdieter Lang, Curvature computation for A247335 and A247512.
Index entries for linear recurrences with constant coefficients, signature (31,-279,729).
FORMULA
One step recurrence: a(n) = 11*a(n-1) - 9^(n-1) + 20*sqrt((a(n-1) - 9^(n-1))*a(n-1)/10), a(0) = 1, n >= 1.
a(n) = (9^n)*(1 + S(n, 22/9) - (11/9)*S(n-1, 22/9))/2, with Chebyshev's S-polynomials (see A049310). For 9^n*S(n, 22/9) see A246645(n). The positive integer sequence sqrt((a(n) - 9^n)*a(n)/10) = A246645(n-1).
O.g.f.: (1 - 21*x + 90*x^2)/((1 - 9*x)*(1 - 22*x + 81*x^2)) = (1/2)*((1 -11*x)/(1 - 22*x + 81*x^2 ) - 1/(1 - 9*x)).
For the proofs see the W. Lang link with a(n) = B'(n).
a(n) = 31*a(n-1)-279*a(n-2)+729*a(n-3). - Colin Barker, Sep 30 2014
MATHEMATICA
CoefficientList[Series[(1 - 21*x + 90*x^2)/((1 - 9*x)*(1 - 22*x + 81*x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{31, -279, 729}, {1, 10, 121}, 50] (* G. C. Greubel, Dec 20 2017 *)
PROG
(PARI) Vec(-(6*x-1)*(15*x-1)/((9*x-1)*(81*x^2-22*x+1)) + O(x^100)) \\ Colin Barker, Sep 30 2014
(Magma) I:=[1, 10, 121]; [n le 3 select I[n] else 31*Self(n-1) - 279*Self(n-2) + 729*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 20 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Sep 30 2014
STATUS
approved