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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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32
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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
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 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, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf
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.
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).
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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] *)
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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 * A179558 A179556 A179477
Adjacent sequences: A004186 A004187 A004188 * A004190 A004191 A004192
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KEYWORD
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easy,nonn
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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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