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A001090 a(n) = 8*a(n-1)-a(n-2); a(0) = 0, a(1) = 1.
(Formerly M4554 N1936)
35
0, 1, 8, 63, 496, 3905, 30744, 242047, 1905632, 15003009, 118118440, 929944511, 7321437648, 57641556673, 453811015736, 3572846569215, 28128961537984, 221458845734657, 1743541804339272, 13726875588979519, 108071462907496880 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of units of a(n) belongs to a periodic sequence: 0, 1, 8, 3, 6, 5, 4, 7, 2, 9. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009

This sequence gives the values of y in solutions of the Diophantine equation x^2 - 15*y^2 = 1; the corresponding values of x are in A001091. - Vincenzo Librandi, Nov 12 2010 [edited by Jon E. Schoenfield, May 02 2014]

For n>= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 8'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

For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,7}. - Milan Janjic, Jan 25 2015

REFERENCES

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009) - From N. J. A. Sloane, May 30 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

H. Brocard, Notes élémentaires sur le problème de Peel, Nouvelle Correspondance Mathématique, 4 (1878), 337-343.

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 231-242.

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=8, 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=10.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (8,-1).

FORMULA

15*a(n)^2 - A001091(n)^2 = -1.

a(n) = S(2*n-1, sqrt(10))/sqrt(10) = S(n-1, 8); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310, with S(-1, x) := 0.

a(n)={{(4+sqrt(15))^n} - {(4-sqrt(15))^n}}/2*sqrt(15). G.f.(x)=x/(1-8x+x^2). - Barry E. Williams, Aug 18 2000

Lim. n-> Inf. a(n)/a(n-1) = 4 + sqrt(15). - Gregory V. Richardson, Oct 13 2002

a(n) = 7*(a(n-1)+a(n-2))-a(n-3). a(n) = 9*(a(n-1)-a(n-2))+a(n-3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Feb 07 2007

[A070997(n-1), a(n)] = [1,6; 1,7]^n * [1,0]. - Gary W. Adamson, Mar 21 2008

a(-n) = -a(n). - Michael Somos, Apr 05 2008

a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*7^k. - Philippe Deléham, Feb 10 2012

Product {n >= 1} (1 + 1/a(n)) = 1/3*(3 + sqrt(15)). - Peter Bala, Dec 23 2012

Product {n >= 2} (1 - 1/a(n)) = 1/8*(3 + sqrt(15)). - Peter Bala, Dec 23 2012

EXAMPLE

G.f. = x + 8*x^2 + 63*x^3 + 496*x^4 + 3905*x^5 + 30744*x^6 + 242047*x^7 + ...

MAPLE

A001090:=1/(1-8*z+z**2); # Simon Plouffe in his 1992 dissertation

MATHEMATICA

lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 4]], {n, 0, 6^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)

LinearRecurrence[{8, -1}, {0, 1}, 30] (* Harvey P. Dale, Aug 29 2012 *)

a[ n_] := ChebyshevU[ n - 1, 4]; (* Michael Somos, May 28 2014 *)

PROG

(PARI) {a(n) = subst(poltchebi(n+1) - 4 * poltchebi(n), x, 4) / 15}; /* Michael Somos, Apr 05 2008 */

(PARI) {a(n) = polchebyshev(n-1, 2, 4)}; /* Michael Somos, May 28 2014 */

(Sage) [lucas_number1(n, 8, 1) for n in range(22)] # Zerinvary Lajos, Jun 25 2008

(Sage) [lucas_number1(n, 8, 1) for n in xrange(0, 22)] # Zerinvary Lajos, Apr 23 2009

CROSSREFS

Cf. A000027, A001906, A001353, A004254, A001109, A004187, A001091.

a(n)=sqrt((A001091(n)^2-1)/15).

Cf. A070997.

Sequence in context: A081107 A164592 A242631 * A243782 A105219 A060071

Adjacent sequences:  A001087 A001088 A001089 * A001091 A001092 A001093

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Wolfdieter Lang, Aug 02 2000

STATUS

approved

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Last modified March 22 20:04 EDT 2017. Contains 283897 sequences.