A057076 A Chebyshev or generalized Fibonacci sequence.

2, 11, 119, 1298, 14159, 154451, 1684802, 18378371, 200477279, 2186871698, 23855111399, 260219353691, 2838557779202, 30963916217531, 337764520613639, 3684445810532498, 40191139395243839, 438418087537149731, 4782407823513403202, 52168067971110285491, 569066339858699737199

Offset: 0

Author

Wolfdieter Lang, Oct 31 2002

Keywords

• nonn
• easy

Comments

From Klaus Purath, Sep 25 2024: (Start)

a(n) = (t(i+3n) - t(i))/(t(i+2n) - t(i+n)) - 1 for n >= 1, where (t) is any sequence satisfying t(i) = 12t(i-1) - 12t(i-2) + t(i-3) or t(i) = 11t(i-1) - t(i-2) without regard to initial values and including this sequence itself, as long as t(i+2n) - t(i+n) != 0 for integer i.

a(n) = (t(i+3n) + t(i))/(t(i+2n) + t(i+n)) + 1 for i >= 0, n >= 1, where (t) is any sequence satisfying t(i) = 10t(i-1) + 10t(i-2) - t(i-3) or t(i) = 11t(i-1) - t(i-2) without regard to initial values and including this sequence itself, as long as t(i+2n) + t(i+n) != 0.

a(n) = (t(i-n) + t(i+n))/t(i) for i >= n >= 0, where (t) is any recurrence of the form (11,-1) including this sequence itself, as long as t(i) != 0.

a(n) = t(n) - t(n-1) = (t(n+1) - t(n-2))/12, where (t) is any third order recurrence with constant coefficients (12,-12,1) and initial values t(0) = x, t(1) = x + 2, t(2) = x + 13 for integer x.

a(n) = t(n-1) + t(n) = (t(n-2) + t(n+1))/10, where (t) is any third order recurrence with constant coefficients (10,10,-1) and initial values t(0) = x, t(1) = 2 - x, t(2) = x + 9 for integer x. (End)

Examples

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..963
• Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• P. Bhadouria, D. Jhala, and B. Singh, Binomial Transforms of the k-Lucas Sequences and its 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).

Crossrefs

Cf. A004190, A049310, A053120.

Programs

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 *)
LinearRecurrence[{11,-1},{2,11},30] (* Harvey P. Dale, Sep 13 2024 *)

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

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

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

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

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

E.g.f.: 2*exp(11*x/2)*cosh(3*sqrt(13)*x/2). - Stefano Spezia, Aug 07 2024

From Klaus Purath, Sep 25 2024: (Start)

a(n) = (a(n-1)*a(n-2) + 1287)/a(n-3) for integer n.

a(n+1)^2 - a(n)*a(n+2) = -117 for integer n. (End)