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A004189 a(n) = 10*a(n-1) - a(n-2); a(0) = 0, a(1) = 1. 55
0, 1, 10, 99, 980, 9701, 96030, 950599, 9409960, 93149001, 922080050, 9127651499, 90354434940, 894416697901, 8853812544070, 87643708742799, 867583274883920, 8588189040096401, 85014307126080090, 841554882220704499 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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, 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

LINKS

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.

D. Birmajer, J. B. Gil, M. D. Weiner, n the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 12.

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

Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.

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:1507.04917 [math.NT], 2015

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (10,-1).

FORMULA

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.

A001079(n) = sqrt{24*[a(n)^2]+1}, that is a(n) = sqrt((A001079(n)^2-1)/24).

From Barry E. Williams, Aug 18 2000: (Start)

a(n) = ( (5+2*sqrt(6))^n - (5-2*sqrt(6))^n )/(4*sqrt(6)).

G.f.: x/(1-10*x+x^2). (End)

a(-n) = -a(n). - Michael Somos, Sep 05 2006

From Mohamed Bouhamida, May 26 2007: (Start)

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). (End)

a(n+1) = Sum_{k=0..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).

EXAMPLE

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 + ...

MAPLE

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

seq( simplify(ChebyshevU(n-1, 5)), n=0..20); # G. C. Greubel, Dec 23 2019

MATHEMATICA

Table[GegenbauerC[n-1, 1, 5], {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008; modified by G. C. Greubel, Jun 06 2019 *)

LinearRecurrence[{10, -1}, {0, 1}, 20] (* Jean-François Alcover, Nov 15 2017 *)

ChebyshevU[Range[21] -2, 5] (* G. C. Greubel, Dec 23 2019 *)

PROG

(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

(PARI) vector(21, n, polchebyshev(n-2, 2, 5) ) \\ G. C. Greubel, Dec 23 2019

(Sage) [lucas_number1(n, 10, 1) for n in range(22)] # Zerinvary Lajos, Jun 25 2008

(Sage) [chebyshev_U(n-1, 5) for n in (0..20)] # G. C. Greubel, Dec 23 2019

(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

(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

CROSSREFS

Cf. A001109, A001353, A001906, A004187, A004254, A018913, A046173.

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).

Cf. A323182.

Sequence in context: A171315 A081109 A242633 * A322054 A179558 A179556

Adjacent sequences:  A004186 A004187 A004188 * A004190 A004191 A004192

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 26 13:49 EDT 2020. Contains 334626 sequences. (Running on oeis4.)