

A057076


A Chebyshev or generalized Fibonacci sequence.


7



2, 11, 119, 1298, 14159, 154451, 1684802, 18378371, 200477279, 2186871698, 23855111399, 260219353691, 2838557779202, 30963916217531, 337764520613639, 3684445810532498, 40191139395243839, 438418087537149731
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OFFSET

0,1


LINKS

Table of n, a(n) for n=0..17.
P. Bhadouria, D. Jhala, B. Singh, Binomial Transforms of the kLucas Sequences and its [sic] Properties, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 8192; sequence R_3.
S. Falcon, Relationships between Some kFibonacci Sequences, Applied Mathematics 5 (2014), 22262234
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(n2, 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.: (211x)/(111x+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 := (11sqrt(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((211*x)/(111*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(n1)^2), n>=1.
Sequence in context: A304639 A130222 A197993 * A251663 A118794 A222879
Adjacent sequences: A057073 A057074 A057075 * A057077 A057078 A057079


KEYWORD

nonn,easy


AUTHOR

Wolfdieter Lang, Oct 31 2002


STATUS

approved



