A246645 Expansion of 1/(1 - 22*x + 81*x^2), used in A246643.

1, 22, 403, 7084, 123205, 2136706, 37027927, 641541208, 11114644489, 192557340910, 3335975296411, 57794311907332, 1001260862952013, 17346399720450394, 300518663950795615, 5206352229561021616, 90197737270328030737, 1562635689352773925318, 27071968446864455867299

Offset: 0

Author

Wolfdieter Lang, Sep 30 2014

Keywords

• nonn
• easy

Comments

This sequence is used in the formula for the curvature in a touching circle problem considered in A247512 and A246643.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..800
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (22,-81).

Crossrefs

Cf. A247512, A246643, A246646.

Programs

Magma

I:=[1, 22]; [n le 2 select I[n] else 22*Self(n-1) - 81*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 20 2017

Mathematica

CoefficientList[Series[1/(1 - 22*x + 81*x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{22,-81}, {1,22}, 50] (* G. C. Greubel, Dec 20 2017 *)

PARI

Vec(1/(1 - 22*x + 81*x^2) + O(x^40)) \\ Michel Marcus, Sep 30 2014

Formula

O.g.f.: 1/(1 - 22*x + 81*x^2).

a(n) = 9^n*S(n, 22/9) with Chebyshev's S-polynomials (see A049310).

a(n) = 22*a(n-1) - 81*a(n-2), n >= 1, a(-1) = 0 and a(0) = 1.

a(n) = 9^n*(ap^(n+1) - am^(n+1))/(ap - am), n >= 1, with ap := (11 + 2*sqrt(10))/9 and am = 1/ap = (11 - 2*sqrt(10))/9 (Binet - de Moivre formula). a(0) = 1 (via L'Hopital's rule).

a(n) = 9^(n+1)*sinh(2*(n + 1)*arccsch(3))/(2*sqrt(10)). - Federico Provvedi, Feb 02 2021