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%I
%S 0,1,10,99,980,9701,96030,950599,9409960,93149001,922080050,
%T 9127651499,90354434940,894416697901,8853812544070,87643708742799,
%U 867583274883920,8588189040096401,85014307126080090,841554882220704499
%N a(n) = 10*a(n-1)-a(n-2); a(0) = 0, a(1) = 1.
%C Indices of square numbers which are also generalized pentagonal numbers.
%C If t(n) denotes the n-th triangular number, t(A105038(n))=a(n)*a(n+1). - Robert Phillips (bobanne(AT)bellsouth.net), May 25 2008
%C The n:th term is a(n)=((5+Sqrt(24))^n-(5-Sqrt(24))^n)/(2*Sqrt(24)) [From Sture Sjoestedt (sture.sjostedt(AT)spray.se), May 31 2009]
%C Number of units of a(n) belongs to a periodic sequence: 0, 1, 0, 9.We conclude that a(n) and a(n+4) have the same number of units. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 05 2009]
%C For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 10's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). [From John M. Campbell, Jul 08 2011]
%D A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=10, q=-1.
%D E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 231-242.
%D D. Fortin, B-SPLINE TOEPLITZ INVERSE UNDER CORNER PERTURBATIONS, International Journal of Pure and Applied Mathematics, Volume 77, No. 1, 2012, 107-118; http://ijpam.eu/contents/2012-77-1/11/11.pdf. - From _N. J. A. Sloane_, Oct 22 2012
%D W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=12.
%H Vincenzo Librandi, <a href="/A004189/b004189.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F a(n) = S(2*n-1, sqrt(12))/sqrt(12) = S(n-1, 10); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(-1, x) := 0.
%F a(n)={[(5+2*sqrt(6))^n - (5-2*sqrt(6))^n]}/4*sqrt(6). G.f.(x)=x/(1-10*x+x^2). - Barry E. Williams, Aug 18 2000
%F G.f.: x/(1-10*x+x^2). a(-n)=-a(n). - Michael Somos Sep 05 2006
%F a(n) = 9*(a(n-1)+a(n-2))-a(n-3), a(n) = 11*(a(n-1)-a(n-2))+a(n-3). a(n)=10*a(n-1)-a(n-2). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 26 2007
%F a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*9^k. - Philippe Deléham, Feb 10 2012
%F Product {n >= 1} (1 + 1/a(n)) = 1/2*(2 + sqrt(6)). - _Peter Bala_, Dec 23 2012
%F Product {n >= 2} (1 - 1/a(n)) = 1/5*(2 + sqrt(6)). - _Peter Bala_, Dec 23 2012
%e a(2)=10 and (3(-8)^2-(-8))/2=10^2, a(3)=99 and (3(81)^2-(81))/2=99^2. - Michael Somos Sep 05 2006
%t lst={};Do[AppendTo[lst, GegenbauerC[n, 1, 5]], {n, 0, 8^2}];lst [From _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008]
%o (PARI) {a(n)=subst(poltchebi(n+1)-5*poltchebi(n), 'x, 5)/24} /* Michael Somos Sep 05 2006 */
%o sage: [lucas_number1(n,10,1) for n in range(22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
%o (MAGMA) [ n eq 1 select 0 else n eq 2 select 1 else 10*Self(n-1)-Self(n-2): n in [1..20] ]; // Vincenzo Librandi, Aug 19 2011
%Y Cf. A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913.
%Y A001079(n) = sqrt{24*[a(n)^2]+1}, that is a(n) = sqrt((A001079(n)^2-1)/24).
%Y A046173(n)=a(2n-1).
%K easy,nonn
%O 0,3
%A _N. J. A. Sloane_.
%E More terms from _James A. Sellers_, Sep 07 2000
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