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

%I A004189 #176 Jan 05 2025 19:51:33
%S A004189 0,1,10,99,980,9701,96030,950599,9409960,93149001,922080050,
%T A004189 9127651499,90354434940,894416697901,8853812544070,87643708742799,
%U A004189 867583274883920,8588189040096401,85014307126080090,841554882220704499,8330534515080964900,82463790268588944501,816307368170808480110
%N A004189 a(n) = 10*a(n-1) - a(n-2); a(0) = 0, a(1) = 1.
%C A004189 Indices of square numbers which are also generalized pentagonal numbers.
%C A004189 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 A004189 The n-th term is a(n) = ((5+sqrt(24))^n - (5-sqrt(24))^n)/(2*sqrt(24)). - _Sture Sjöstedt_, May 31 2009
%C A004189 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 A004189 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 A004189 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 A004189 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
%C A004189 All possible solutions for y in Pell equation x^2 - 24*y^2 = 1. The values for x are given in A001079. - _Herbert Kociemba_, Jun 05 2022
%C A004189 Dickson on page 384 gives the Diophantine equation "(20)  24x^2 + 1 = y^2" and later states "... three consecutive sets (x_i, y_i) of solutions of (20) or 2x^2 + 1 = 3y^2 satisfy x_{n+1} = 10x_n - x_{n-1}, y_{n+1} = 10y_n - y_{n-1} with (x_1, y_1) = (0, 1) or (1, 1), (x_2, y_2) = (1, 5) or (11, 9), respectively." The first set of values (x_n, y_n) = (A001079(n-1), a(n-1)). - _Michael Somos_, Jun 19 2023
%D A004189 L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, p. 384.
%H A004189 Vincenzo Librandi, <a href="/A004189/b004189.txt">Table of n, a(n) for n = 0..1000</a>
%H A004189 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 A004189 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 A004189 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 A004189 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 A004189 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=10, q=-1.
%H A004189 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 A004189 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A004189 Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, <a href="https://arxiv.org/abs/1904.13002">The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d))</a>, arXiv:1904.13002 [math.NT], 2019.
%H A004189 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. 38,5 (2000) 408-419; Eq.(44), lhs, m=12.
%H A004189 Roger B. Nelson, <a href="http://www.jstor.org/stable/10.4169/math.mag.89.3.159">Multi-Polygonal Numbers</a>, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.
%H A004189 Robert Phillips, <a href="https://web.archive.org/web/20100713033314/http://www.usca.edu/math/~mathdept/bobp/pdf/polgonal.pdf">Polynomials of the form 1+4ke+4ke^2</a>, 2008.
%H A004189 Wikipedia, <a href="https://en.wikipedia.org/wiki/Curse_of_39">Curse of 39</a>
%H A004189 Jianqiang Zhao, <a href="http://arxiv.org/abs/1507.04917">Finite Multiple zeta Values and Finite Euler Sums</a>, arXiv:1507.04917 [math.NT], 2015.
%H A004189 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H A004189 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-1).
%F A004189 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 A004189 A001079(n) = sqrt(24*(a(n)^2)+1), that is a(n) = sqrt((A001079(n)^2-1)/24).
%F A004189 From _Barry E. Williams_, Aug 18 2000: (Start)
%F A004189 a(n) = ( (5+2*sqrt(6))^n - (5-2*sqrt(6))^n )/(4*sqrt(6)).
%F A004189 G.f.: x/(1-10*x+x^2). (End)
%F A004189 a(-n) = -a(n). - _Michael Somos_, Sep 05 2006
%F A004189 From _Mohamed Bouhamida_, May 26 2007: (Start)
%F A004189 a(n) = 9*(a(n-1) + a(n-2)) - a(n-3).
%F A004189 a(n) = 11*(a(n-1) - a(n-2)) + a(n-3).
%F A004189 a(n) = 10*a(n-1) - a(n-2). (End)
%F A004189 a(n+1) = Sum_{k=0..n} A101950(n,k)*9^k. - _Philippe Deléham_, Feb 10 2012
%F A004189 From _Peter Bala_, Dec 23 2012: (Start)
%F A004189 Product {n >= 1} (1 + 1/a(n)) = 1/2*(2 + sqrt(6)).
%F A004189 Product {n >= 2} (1 - 1/a(n)) = 1/5*(2 + sqrt(6)). (End)
%F A004189 a(n) = (A054320(n-1) + A072256(n))/2. - _Richard R. Forberg_, Nov 21 2013
%F A004189 a(2*n - 1) = A046173(n).
%F A004189 E.g.f.: exp(5*x)*sinh(2*sqrt(6)*x)/(2*sqrt(6)). - _Stefano Spezia_, Dec 12 2022
%F A004189 a(n) = Sum_{k = 0..n-1} binomial(n+k, 2*k+1)*8^k = Sum_{k = 0..n-1} (-1)^(n+k+1)* binomial(n+k, 2*k+1)*12^k. - _Peter Bala_, Jul 18 2023
%e A004189 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 A004189 G.f. = x + 10*x^2 + 99*x^3 + 980*x^4 + 9701*x^5 + 96030*x^6 + ...
%p A004189 A004189 := proc(n)
%p A004189     option remember;
%p A004189     if n <= 1 then
%p A004189         n ;
%p A004189     else
%p A004189         10*procname(n-1)-procname(n-2) ;
%p A004189     end if;
%p A004189 end proc:
%p A004189 seq(A004189(n),n=0..20) ; # _R. J. Mathar_, Apr 30 2017
%p A004189 seq( simplify(ChebyshevU(n-1, 5)), n=0..20); # _G. C. Greubel_, Dec 23 2019
%t A004189 Table[GegenbauerC[n-1,1,5], {n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008; modified by _G. C. Greubel_, Jun 06 2019 *)
%t A004189 LinearRecurrence[{10, -1}, {0, 1}, 20] (* _Jean-François Alcover_, Nov 15 2017 *)
%t A004189 ChebyshevU[Range[21] -2, 5] (* _G. C. Greubel_, Dec 23 2019 *)
%o A004189 (PARI) {a(n) = subst(poltchebi(n+1) - 5*poltchebi(n), 'x, 5) / 24}; /* _Michael Somos_, Sep 05 2006 */
%o A004189 (PARI) a(n)=([9,1;8,1]^(n-1)*[1;1])[1,1] \\ _Charles R Greathouse IV_, Jan 14 2016
%o A004189 (PARI) vector(21, n, n--; polchebyshev(n-1, 2, 5) ) \\ _G. C. Greubel_, Dec 23 2019
%o A004189 (Sage) [lucas_number1(n,10,1) for n in range(22)] # _Zerinvary Lajos_, Jun 25 2008
%o A004189 (Sage) [chebyshev_U(n-1,5) for n in (0..20)] # _G. C. Greubel_, Dec 23 2019
%o A004189 (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
%o A004189 (GAP) m:=5;; a:=[0,1];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Dec 23 2019
%Y A004189 Cf. A001079, A001109, A001353, A001906, A004187, A004254, A018913, A046173, A049310, A054320, A072256, A101950, A105138, A108741 (squares).
%Y A004189 Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), this sequence (m=5), A004191 (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
%Y A004189 Cf. A323182.
%K A004189 easy,nonn
%O A004189 0,3
%A A004189 _N. J. A. Sloane_