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a(n) = 9*a(n - 1) - a(n - 2) for n>1, a(0)=0, a(1)=1.
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%I #100 Nov 16 2023 13:38:27

%S 0,1,9,80,711,6319,56160,499121,4435929,39424240,350382231,3114015839,

%T 27675760320,245967827041,2186034683049,19428344320400,

%U 172669064200551,1534593233484559,13638670037160480,121213437100959761

%N a(n) = 9*a(n - 1) - a(n - 2) for n>1, a(0)=0, a(1)=1.

%C Define the sequence L(a_0,a_1) by a_{n+2} is the greatest integer such that a_{n+2}/a_{n+1}<a_{n+1}/a_n for n >= 0. This is L(1,9).

%C For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 9'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 For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,8}. - _Milan Janjic_, Jan 25 2015

%C Not to be confused with the Pisot L(1,9) sequence, which is A001019. - _R. J. Mathar_, Feb 13 2016

%C Lim_{n->oo} a(n+1)/a(n) = (9 + sqrt(77))/2 = A092290 + 1 = 8.887482... - _Wolfdieter Lang_, Nov 16 2023

%H Vincenzo Librandi, <a href="/A018913/b018913.txt">Table of n, a(n) for n = 0..1000</a>

%H Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.

%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 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 D. W. Boyd, <a href="http://www.researchgate.net/publication/258834801">Linear recurrence relations for some generalized Pisot sequences</a>, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.

%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=9, 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=11.

%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 (9,-1).

%H <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>

%F G.f.: x/(1-9*x+x^2).

%F a(n) = S(2*n-1, sqrt(11))/sqrt(11) = S(n-1, 9); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(-1, x) := 0.

%F a(n) = (((9+sqrt(77))/2)^n - ((9-sqrt(77))/2)^n)/sqrt(77). - _Barry E. Williams_, Aug 21 2000

%F a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*8^k. - _Philippe Deléham_, Feb 10 2012

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

%F Product {n >= 1} (1 + 1/a(n)) = 1/7*(7 + sqrt(77)).

%F Product {n >= 2} (1 - 1/a(n)) = 1/18*(7 + sqrt(77)). (End)

%F a(n) = Sum_{k = 0..n-1} binomial(n+k, 2*k+1)*7^k = Sum_{k = 0..n-1} (-1)^(n+k+1)* binomial(n+k, 2*k+1)*11^k. - _Peter Bala_, Jul 17 2023

%e G.f. = x + 9*x^2 + 80*x^3 + 711*x^4 + 6319*x^5 + 56160*x^6 + 499121*x^7 + ...

%t CoefficientList[Series[x/(1 - 9*x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 23 2012 *)

%o (Sage) [lucas_number1(n,9,1) for n in range(22)] # _Zerinvary Lajos_, Jun 25 2008

%o (Magma) I:=[0, 1]; [n le 2 select I[n] else 9*Self(n-1) - Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Dec 23 2012

%o (PARI) concat(0, Vec(x/(1-9*x+x^2) + O(x^30))) \\ _Michel Marcus_, Sep 06 2017

%Y Cf. A000027, A001906, A001353, A004254, A001109, A004187, A001090.

%Y Cf. A056918(n)=sqrt{77*(a(n))^2 +4}, that is, a(n)=sqrt((A056918(n)^2 - 4)/77).

%Y Cf. A092290 + 1.

%K nonn,easy

%O 0,3

%A _R. K. Guy_

%E G.f. adapted to the offset by _Vincenzo Librandi_, Dec 23 2012