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%I #72 Dec 31 2023 10:12:21
%S 1,8,71,631,5608,49841,442961,3936808,34988311,310957991,2763633608,
%T 24561744481,218292066721,1940066856008,17242309637351,
%U 153240719880151,1361924169284008,12104076803675921,107574767063799281,956068826770517608
%N a(n) = 9*a(n-1) - a(n-2) for n > 0, a(0)=1, a(-1)=1.
%C A Pellian sequence.
%C In general, Sum_{k=0..n} binomial(2n-k,k)j^(n-k) = (-1)^n*U(2n, i*sqrt(j)/2), i=sqrt(-1). - _Paul Barry_, Mar 13 2005
%C a(n) = L(n,9), where L is defined as in A108299; see also A057081 for L(n,-9). - _Reinhard Zumkeller_, Jun 01 2005
%C Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7,8} which do not end in 0. - _Tanya Khovanova_, Jan 10 2007
%C For positive n, a(n) equals the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(7)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - _John M. Campbell_, Jul 08 2011
%C Positive values of x (or y) satisfying x^2 - 9xy + y^2 + 7 = 0. - _Colin Barker_, Feb 09 2014
%H Vincenzo Librandi, <a href="/A070998/b070998.txt">Table of n, a(n) for n = 0..200</a>
%H Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-1).
%F a(n) ~ (1/11)*sqrt(11)*((1/2)*(sqrt(11) + sqrt(7)))^(2*n+1).
%F Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then q(n, 7) = a(n). - _Benoit Cloitre_, Nov 10 2002
%F a(n)*a(n+3) = 63 + a(n+1)*a(n+2). - _Ralf Stephan_, May 29 2004
%F a(n) = (-1)^n*U(2n, i*sqrt(7)/2), U(n, x) Chebyshev polynomial of second kind, i=sqrt(-1). - _Paul Barry_, Mar 13 2005
%F G.f.: (1-x)/(1-9*x+x^2). - _Philippe Deléham_, Nov 03 2008
%F a(n) = A018913(n+1) - A018913(n). - _R. J. Mathar_, Jun 07 2013
%t CoefficientList[Series[(1 - x)/(1 - 9 x + x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Feb 10 2014 *)
%t LinearRecurrence[{9,-1},{1,8},30] (* _Harvey P. Dale_, Sep 24 2015 *)
%o (Sage) [lucas_number1(n, 9, 1) - lucas_number1(n-1, 9, 1) for n in range(1, 19)] # _Zerinvary Lajos_, Nov 10 2009
%o (Magma) I:=[1,8]; [n le 2 select I[n] else 9*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 10 2014
%Y Cf. A057081, A056918.
%Y Row 9 of array A094954.
%Y Cf. similar sequences listed in A238379.
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org), May 18 2002
%E More terms from _Vincenzo Librandi_, Feb 10 2014
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