%I #103 Aug 07 2024 15:41:57

%S 1,11,120,1309,14279,155760,1699081,18534131,202176360,2205405829,

%T 24057287759,262424759520,2862615066961,31226340977051,

%U 340627135680600,3715672151509549,40531766530924439,442133759688659280,4822939590044327641,52610201730798944771,573889279448744064840

%N Expansion of 1/(1 - 11*x + x^2).

%C Chebyshev or generalized Fibonacci sequence.

%C This is the m=13 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..12 (nonnegative) sequences are A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913 and A004189. The m=1..3 (signed) sequences are A049347, A056594, A010892.

%C All positive integer solutions of Pell equation b(n)^2 - 117*a(n)^2 = +4 together with b(n+1)=A057076(n+1), n >= 0. - _Wolfdieter Lang_, Aug 31 2004

%C For positive n, a(n) equals the permanent of the tridiagonal matrix of order n with 11's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011

%C a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,10}. - _Milan Janjic_, Jan 25 2015

%H Vincenzo Librandi, <a href="/A004190/b004190.txt">Table of n, a(n) for n = 0..900</a>

%H K. Andersen, L. Carbone, and D. Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.

%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 D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, example 12

%H S. Falcon, <a href="http://journal.ccsenet.org/index.php/jmr/article/viewFile/14516/10822">Generalized Fibonacci Sequences Generated from a k-Fibonacci Sequence</a>, Journal of Mathematics Research Vol. 4, No. 2; April 2012. - From _N. J. A. Sloane_, Sep 22 2012

%H R. Flórez, R. A. Higuita, and A. Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=11, q=-1.

%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

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

%H Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=13.

%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 Recursion: a(n) = 11*a(n-1) - a(n-2), n >= 1; a(-1)=0, a(0)=1.

%F a(n) = S(2*n+1, sqrt(13))/sqrt(13) = S(n, 11); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.

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

%F a(n) = ((11+3*sqrt(13))^(n+1) - (11-3*sqrt(13))^(n+1))/(2^(n+1)*3*sqrt(13)). - _Rolf Pleisch_, May 22 2011

%F a(n) = Sum_{k=0..n} A101950(n,k)*10^k. - _Philippe Deléham_, Feb 10 2012

%F From _Peter Bala_, Dec 23 2012: (Start)

%F Product_{n>=0} (1 + 1/a(n)) = 1/3*(3 + sqrt(13)).

%F Product_{n>=1} (1 - 1/a(n)) = 3/22*(3 + sqrt(13)). (End)

%F a(n) = sqrt((A057076(n+1)^2 - 4)/117).

%F a(n) = A075835(n+1)/3 = A006190(2*n+2)/3. - _Vladimir Reshetnikov_, Sep 16 2016

%F a(n) = -a(-2-n) for all n in Z. - _Michael Somos_, Jul 14 2018

%F E.g.f.: exp(11*x/2)*(39*cosh(3*sqrt(13)*x/2) + 11*sqrt(13)*sinh(3*sqrt(13)*x/2))/39. - _Stefano Spezia_, Aug 07 2024

%e G.f. = 1 + 11*x + 120*x^2 + 1309*x^3 + 14279*x^4 + 155760*x^5 + ...

%p with(combinat):seq(fibonacci(2*n+2, 3)/3, n=0..20); # _Zerinvary Lajos_, Apr 20 2008

%t Join[{a=1,b=11},Table[c=11*b-a; a=b; b=c, {n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 20 2011 *)

%t CoefficientList[Series[1/(1-11*x+x^2),{x,0,30}],x] (* _Vincenzo Librandi_, Jun 13 2012 *)

%t Table[Fibonacci[2n + 2, 3]/3, {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)

%t a[ n_] := ChebyshevU[n, 11/2]; (* _Michael Somos_, Jul 14 2018 *)

%o (Sage) [lucas_number1(n,11,1) for n in range(1, 20)] # _Zerinvary Lajos_, Jun 25 2008

%o (PARI) Vec(1/(1-11*x+x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012

%o (PARI) {a(n) = polchebyshev(n, 2, 11/2)}; /* _Michael Somos_, Jul 14 2018 */

%Y Cf. A000027, A001090, A001109, A001353, A001906, A004187, A004189, A004254, A006190, A010892, A018913, A049310, A049347, A056594, A057076, A075835, A101950.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E _Wolfdieter Lang_, Oct 31 2002