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A057076 A Chebyshev or generalized Fibonacci sequence. 6
2, 11, 119, 1298, 14159, 154451, 1684802, 18378371, 200477279, 2186871698, 23855111399, 260219353691, 2838557779202, 30963916217531, 337764520613639, 3684445810532498, 40191139395243839, 438418087537149731 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Table of n, a(n) for n=0..17.

P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the k-Lucas Sequences and its [sic] Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence R_3.

S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics 5 (2014), 2226-2234

Tanya Khovanova, Recursive Sequences

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (11,-1).

FORMULA

a(n) = S(n, 11) - S(n-2, 11) = 2*T(n, 11/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 11)=A004190(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.

G.f.: (2-11x)/(1-11x+x^2).

a(n) = a(-n). - Michael Somos, Apr 25 2003

a(n) = ap^n + am^n, with ap := (11+sqrt(117))/2 and am := (11-sqrt(117))/2.

EXAMPLE

G.f. = 2 + 11*x +119*x^2 + 1298*x^3 + 14159*x^4 + 154451*x^5 + ...

MATHEMATICA

a[0] = 2; a[1] = 11; a[n_] := 11a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 17}] (* Robert G. Wilson v, Jan 30 2004 *)

a[ n_] := 2 ChebyshevT[ n, 11/2]; (* Michael Somos, May 28 2014 *)

PROG

(PARI) {a(n) = subst( poltchebi(n), x, 11/2) * 2};

(PARI) {a(n) = 2 * poltchebyshev(n, 1, 11/2)}; /* Michael Somos, May 28 2014 */

(PARI) Vec((2-11*x)/(1-11*x+x^2) + O(x^40)) \\ Michel Marcus, Feb 18 2016

(Sage) [lucas_number2(n, 11, 1) for n in range(27)] # Zerinvary Lajos, Jun 25 2008

CROSSREFS

a(n) = sqrt(4+117*A004190(n-1)^2), n>=1.

Sequence in context: A269082 A130222 A197993 * A251663 A118794 A222879

Adjacent sequences:  A057073 A057074 A057075 * A057077 A057078 A057079

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Oct 31 2002

STATUS

approved

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Last modified June 22 07:47 EDT 2017. Contains 288605 sequences.