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A018913 a(n) = 9*a(n - 1) - a(n - 2) for n>1, a(0)=0, a(1)=1.

%I A018913 #112 Jun 22 2025 00:16:53
%S A018913 0,1,9,80,711,6319,56160,499121,4435929,39424240,350382231,3114015839,
%T A018913 27675760320,245967827041,2186034683049,19428344320400,
%U A018913 172669064200551,1534593233484559,13638670037160480,121213437100959761
%N A018913 a(n) = 9*a(n - 1) - a(n - 2) for n>1, a(0)=0, a(1)=1.
%C A018913 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 A018913 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 A018913 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 A018913 Not to be confused with the Pisot L(1,9) sequence, which is A001019. - _R. J. Mathar_, Feb 13 2016
%C A018913 Lim_{n->oo} a(n+1)/a(n) = (9 + sqrt(77))/2 = A092290 + 1 = 8.887482... - _Wolfdieter Lang_, Nov 16 2023
%H A018913 Vincenzo Librandi, <a href="/A018913/b018913.txt">Table of n, a(n) for n = 0..1000</a>
%H A018913 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 A018913 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 A018913 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 A018913 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 A018913 E. I. Emerson, <a href="https://web.archive.org/web/2024*/https://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 A018913 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://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 A018913 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 A018913 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A018913 Wolfdieter Lang, <a href="https://web.archive.org/web/2024*/https://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 A018913 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A018913 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-1).
%H A018913 <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>
%F A018913 G.f.: x/(1-9*x+x^2).
%F A018913 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 A018913 a(n) = (((9+sqrt(77))/2)^n - ((9-sqrt(77))/2)^n)/sqrt(77). - _Barry E. Williams_, Aug 21 2000
%F A018913 a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*8^k. - _Philippe Deléham_, Feb 10 2012
%F A018913 From _Peter Bala_, Dec 23 2012: (Start)
%F A018913 Product {n >= 1} (1 + 1/a(n)) = 1/7*(7 + sqrt(77)).
%F A018913 Product {n >= 2} (1 - 1/a(n)) = 1/18*(7 + sqrt(77)). (End)
%F A018913 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
%F A018913 E.g.f.: 2*exp(9*x/2)*sinh(sqrt(77)*x/2)/sqrt(77). - _Stefano Spezia_, Feb 23 2025
%F A018913 Product_{n >= 1} (a(2*n) + 1)/(a(2*n) - 1) = sqrt(11/7) [telescoping product: ((a(2*n) + 1)/(a(2*n) - 1))^2 = (11 - 4/(a(n+1) - a(n))^2)/(11 - 4/(a(n) - a(n-1))^2), leading to 11 - 7*Product_{k = 1..n} ((a(2*k) + 1)/(a(2*k) - 1))^2 = 4/A070998(n)^2]. - _Peter Bala_, May 18 2025
%e A018913 G.f. = x + 9*x^2 + 80*x^3 + 711*x^4 + 6319*x^5 + 56160*x^6 + 499121*x^7 + ...
%t A018913 CoefficientList[Series[x/(1 - 9*x + x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 23 2012 *)
%o A018913 (Sage) [lucas_number1(n,9,1) for n in range(22)] # _Zerinvary Lajos_, Jun 25 2008
%o A018913 (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 A018913 (PARI) concat(0, Vec(x/(1-9*x+x^2) + O(x^30))) \\ _Michel Marcus_, Sep 06 2017
%Y A018913 Cf. A000027, A001906, A001353, A004254, A001109, A004187, A001090.
%Y A018913 Cf. A056918(n)=sqrt{77*(a(n))^2 +4}, that is, a(n)=sqrt((A056918(n)^2 - 4)/77).
%Y A018913 Cf. A092290 + 1.
%K A018913 nonn,easy
%O A018913 0,3
%A A018913 _R. K. Guy_
%E A018913 G.f. adapted to the offset by _Vincenzo Librandi_, Dec 23 2012