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A246645 Expansion of 1/(1 - 22*x + 81*x^2), used in A246643. 5
1, 22, 403, 7084, 123205, 2136706, 37027927, 641541208, 11114644489, 192557340910, 3335975296411, 57794311907332, 1001260862952013, 17346399720450394, 300518663950795615, 5206352229561021616, 90197737270328030737, 1562635689352773925318, 27071968446864455867299 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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

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 *)

PROG

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

(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

CROSSREFS

Cf. A247512, A246643, A246646.

Sequence in context: A268947 A159761 A049663 * A183539 A269540 A095219

Adjacent sequences:  A246642 A246643 A246644 * A246646 A246647 A246648

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 30 2014

STATUS

approved

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Last modified July 25 16:39 EDT 2021. Contains 346291 sequences. (Running on oeis4.)