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A004189
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a(n) = 10*a(n-1) - a(n-2); a(0) = 0, a(1) = 1.
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39
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0, 1, 10, 99, 980, 9701, 96030, 950599, 9409960, 93149001, 922080050, 9127651499, 90354434940, 894416697901, 8853812544070, 87643708742799, 867583274883920, 8588189040096401, 85014307126080090, 841554882220704499
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OFFSET
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0,3
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COMMENTS
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Indices of square numbers which are also generalized pentagonal numbers.
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
The n-th term is a(n)=((5+sqrt(24))^n-(5-sqrt(24))^n)/(2*sqrt(24)). - Sture Sjöstedt, May 31 2009
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
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
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
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
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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
D. Fortin, B-spline Toeplitz inverse under corner perturbations, International Journal of Pure and Applied Mathematics, Volume 77, No. 1, 2012, 107-118. - From N. J. A. Sloane, Oct 22 2012
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.
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
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.
Roger B. Nelson, Multi-Polygonal Numbers, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.
Wikipedia, Curse of 39
Jianqiang Zhao, Finite Multiple zeta Values and Finite Euler Sums, arXiv preprint arXiv:1507.04917, 2015
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (10,-1).
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FORMULA
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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.
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
a(-n) = -a(n). - Michael Somos, Sep 05 2006
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
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*9^k. - Philippe Deléham, Feb 10 2012
Product {n >= 1} (1 + 1/a(n)) = 1/2*(2 + sqrt(6)). - Peter Bala, Dec 23 2012
Product {n >= 2} (1 - 1/a(n)) = 1/5*(2 + sqrt(6)). - Peter Bala, Dec 23 2012
a(n) = (A054320(n-1) + A072256(n))/2. - Richard R. Forberg, Nov 21 2013
a(2*n - 1) = A046173(n).
G.f. = x/(1-10*x+x^2). - N. J. A. Sloane, Dec 20 2018
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EXAMPLE
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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
G.f. = x + 10*x^2 + 99*x^3 + 980*x^4 + 9701*x^5 + 96030*x^6 + ...
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MAPLE
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A004189 := proc(n)
option remember;
if n <= 1 then
n ;
else
10*procname(n-1)-procname(n-2) ;
end if;
end proc:
seq(A004189(n), n=0..20) ; # R. J. Mathar, Apr 30 2017
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MATHEMATICA
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lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 5]], {n, 0, 8^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008] *)
LinearRecurrence[{10, -1}, {0, 1}, 20] (* Jean-François Alcover, Nov 15 2017 *)
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PROG
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(PARI) {a(n) = subst(poltchebi(n+1) - 5*poltchebi(n), 'x, 5) / 24}; /* Michael Somos, Sep 05 2006 */
(PARI) a(n)=([9, 1; 8, 1]^(n-1)*[1; 1])[1, 1] \\ Charles R Greathouse IV, Jan 14 2016
(Sage) [lucas_number1(n, 10, 1) for n in range(22)] # Zerinvary Lajos, Jun 25 2008
(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
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CROSSREFS
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Cf. A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913.
A001079(n) = sqrt{24*[a(n)^2]+1}, that is a(n) = sqrt((A001079(n)^2-1)/24).
Cf. A046173.
Sequence in context: A171315 A081109 A242633 * A322054 A179558 A179556
Adjacent sequences: A004186 A004187 A004188 * A004190 A004191 A004192
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KEYWORD
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easy,nonn,changed
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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