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A004189 a(n) = 10*a(n-1) - a(n-2); a(0) = 0, a(1) = 1. 32

%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)). - _Sture Sjöstedt_, 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. - 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). - _John M. Campbell_, Jul 08 2011

%C a(n) and b(n) (A001079) are the nonnegative proper solutions of the Pell equation b(n)^2 - 6*(2*a(n))^2 = +1. See the cross reference to A001079 below. - _Wolfdieter Lang_, Jun 26 2013

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

%C For n > 1, this also gives the number of (n-1)-decimal digit numbers which avoid a particular two-digit number with distinct digits. For example, there are a(5) = 9701 4-digit numbers which do not include "39" as a substring; see Wikipedia link. - _Charles R Greathouse IV_, Jan 14 2016

%H Vincenzo Librandi, <a href="/A004189/b004189.txt">Table of n, a(n) for n = 0..1000</a>

%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), pps. 231-242.

%H D. Fortin, <a href="http://ijpam.eu/contents/2012-77-1/11/11.pdf">B-spline Toeplitz inverse under corner perturbations</a>, International Journal of Pure and Applied Mathematics, Volume 77, No. 1, 2012, 107-118. - From _N. J. A. Sloane_, Oct 22 2012

%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=10, q=-1.

%H M. Janjic, <a href="http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf

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

%H W. 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. 38,5 (2000) 408-419; Eq.(44), lhs, m=12.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Curse_of_39">Curse of 39</a>

%H Jianqiang Zhao, <a href="http://arxiv.org/abs/1507.04917">Finite Multiple zeta Values and Finite Euler Sums</a>, arXiv preprint arXiv:1507.04917, 2015

%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 (10,-1).

%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 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

%F a(n) = (A054320(n-1) + A072256(n))/2. - _Richard R. Forberg_, Nov 21 2013

%F a(2*n - 1) = A046173(n).

%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

%e G.f. = x + 10*x^2 + 99*x^3 + 980*x^4 + 9701*x^5 + 96030*x^6 + ...

%t lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 5]], {n, 0, 8^2}]; lst (* _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 (PARI) a(n)=([9,1;8,1]^(n-1)*[1;1])[1,1] \\ _Charles R Greathouse IV_, Jan 14 2016

%o (Sage) [lucas_number1(n,10,1) for n in range(22)] # _Zerinvary Lajos_, 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 Cf. A046173.

%K easy,nonn

%O 0,3

%A _N. J. A. Sloane_

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Last modified December 7 19:05 EST 2016. Contains 278895 sequences.