A057076 - OEIS

%I #50 Oct 18 2024 20:34:19

%S 2,11,119,1298,14159,154451,1684802,18378371,200477279,2186871698,

%T 23855111399,260219353691,2838557779202,30963916217531,

%U 337764520613639,3684445810532498,40191139395243839,438418087537149731,4782407823513403202,52168067971110285491,569066339858699737199

%N A Chebyshev or generalized Fibonacci sequence.

%C From _Klaus Purath_, Sep 25 2024: (Start)

%C 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.

%C 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.

%C 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.

%C 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.

%C 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)

%H Michael De Vlieger, <a href="/A057076/b057076.txt">Table of n, a(n) for n = 0..963</a>

%H Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

%H P. Bhadouria, D. Jhala, and B. Singh, <a href="http://dx.doi.org/10.22436/jmcs.08.01.07">Binomial Transforms of the k-Lucas Sequences and its Properties</a>, The Journal of Mathematics and Computer Science (JMCS), Volume 8, Issue 1, Pages 81-92; sequence R_3.

%H S. Falcon, <a href="http://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics 5 (2014), 2226-2234.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rea#recur1">Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-1).

%F 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.

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

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

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

%F a(n) = a(-n). - _Michael Somos_, Apr 25 2003

%F E.g.f.: 2*exp(11*x/2)*cosh(3*sqrt(13)*x/2). - _Stefano Spezia_, Aug 07 2024

%F From _Klaus Purath_, Sep 25 2024: (Start)

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

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

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

%t 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 *)

%t a[ n_] := 2 ChebyshevT[ n, 11/2]; (* _Michael Somos_, May 28 2014 *)

%t LinearRecurrence[{11,-1},{2,11},30] (* _Harvey P. Dale_, Sep 13 2024 *)

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

%o (PARI) {a(n) = 2 * poltchebyshev(n, 1, 11/2)}; /* _Michael Somos_, May 28 2014 */

%o (PARI) Vec((2-11*x)/(1-11*x+x^2) + O(x^40)) \\ _Michel Marcus_, Feb 18 2016

%o (Sage) [lucas_number2(n,11,1) for n in range(27)] # _Zerinvary Lajos_, Jun 25 2008

%Y Cf. A004190, A049310, A053120.

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Oct 31 2002