login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001079 a(n) = 10*a(n-1) - a(n-2); a(0) = 1, a(1) = 5.
(Formerly M4005 N1659)
45
1, 5, 49, 485, 4801, 47525, 470449, 4656965, 46099201, 456335045, 4517251249, 44716177445, 442644523201, 4381729054565, 43374646022449, 429364731169925, 4250272665676801, 42073361925598085, 416483346590304049 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where r=sqrt(6). - Benoit Cloitre, Feb 14 2004

Appears to give all solutions >1 to the equation x^2=ceiling(x*r*floor(x/r)) where r=sqrt(6). - Benoit Cloitre, Feb 24 2004

(sqrt(2)+sqrt(3))^(2*n)=a(n)+A001078(n)*sqrt(6). - Reinhard Zumkeller, Mar 12 2008

a(n) and b(n) (A004189) are the nonnegative proper solutions to the Pell equation a(n)^2 - 6*(2*b(n))^2 = +1, n >= 0. The formula given below by Gregory V. Richardson follows. - Wolfdieter Lang, Jun 26 2013

a(n) are the integer square roots of (A032528 + 1). They are also the values of m where (A032528(m) - 1) has integer square roots. See A122653 for the integer square roots of (A032528 - 1), and see A122652 for the values of m where (A032528(m) + 1) has integer square roots. - Richard R. Forberg, Aug 05 2013

a(n) are also the values of m where floor(2m^2/3) has integer square roots, excluding m = 0. The corresponding integer square roots are given by A122652(n). - Richard R. Forberg, Nov 21 2013

Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 24 = 0. - Colin Barker, Feb 09 2014

REFERENCES

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

L. Euler, (E388) Vollstaendige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 374.

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).

V. Thébault, Les Récréations Mathématiques. Gauthier-Villars, Paris, 1952, p. 281.

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.

L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.

L. Euler, De solutione problematum diophanteorum per numeros integros, par. 18

Tanya Khovanova, Recursive Sequences

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

Index entries for sequences related to Chebyshev polynomials.

FORMULA

For all members x of the sequence, 6*x^2 -6 is a square. Lim. n-> Inf. a(n)/a(n-1) = 5 + 2*Sqrt(6). - Gregory V. Richardson, Oct 13 2002

a(n) = T(n, 5) = (S(n, 10)-S(n-2, 10))/2 with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(n, 10)= A004189(n+1).

a(n) = sqrt(1+24*A004189(n)^2) (cf. Richardson comment).

a(n)a(n+3) - a(n+1)a(n+2) = 240. - Ralf Stephan, Jun 06 2005

Chebyshev's polynomials T(n,x) evaluated at x=5.

G.f.: (1-5*x)/(1-10*x+x^2). Simon Plouffe in his 1992 dissertation

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

a(-n) = a(n).

a(n+1) = 5*a(n)+2*(6*a(n)^2-6)^(1/2) - Richard Choulet, Sep 19 2007

a(n+1) = 2*A054320(n) + 3*A138288(n). - Reinhard Zumkeller, Mar 12 2008

a(n) = cosh(2*n* arcsinh(sqrt(2))). - Herbert Kociemba, Apr 24 2008

a(n) = (-1)^n * cos(2*n* arcsin(sqrt(3))). - Artur Jasinski, Oct 29 2008

a(n) = cos(2*n* arccos(sqrt(3))). - Artur Jasinski, Sep 10 2016

A001079(n) = 142238(2n-1) = A041006(2n-1) = A041038(2n-1), for all n > 0. - M. F. Hasler, Feb 14 2009

2*a(n)^2 = 3*A122652(n)^2 + 2. - Charlie Marion, Feb 01 2013

E.g.f.: cosh(2*sqrt(6)*x)*exp(5*x). - Ilya Gutkovskiy, Sep 10 2016

EXAMPLE

Pell equation: n = 0: 1^2 - 24*0^2 = +1, n = 1: 5^2 - 6*(1*2)^2 = 1, n = 2: 49^2 - 6*(2*10)^2 = +1. - Wolfdieter Lang, Jun 26 2013

G.f. = 1 + 5*x + 49*x^2 + 485*x^3 + 4801*x^4 + 47525*x^5 + 470449*x^6 + ...

MAPLE

A001079 := proc(n)

    option remember;

    if n <= 1 then

        op(n+1, [1, 5]) ;

    else

        10*procname(n-1)-procname(n-2) ;

    end if;

end proc:

seq(A001079(n), n=0..20) ; # R. J. Mathar, Apr 30 2017

MATHEMATICA

Table[(-1)^n Round[N[Cos[2 n ArcSin[Sqrt[3]]], 50]], {n, 0, 20}] (* Artur Jasinski, Oct 29 2008 *)

a[ n_] := ChebyshevT[n, 5]; (* Michael Somos, Aug 24 2014 *)

PROG

(PARI) {a(n) = subst(poltchebi(n), 'x, 5)}; /* Michael Somos, Sep 05 2006 */

(PARI) {a(n) = real((5 + 2*quadgen(24))^n)}; /* Michael Somos, Sep 05 2006 */

(PARI) {a(n) = n = abs(n); polsym(1 - 10*x + x^2, n)[n+1] / 2}; /* Michael Somos, Sep 05 2006 */

(MAGMA) I:=[1, 5]; [n le 2 select I[n] else 10*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 10 2016

CROSSREFS

Cf. A004189, A001078, A046173, A046172, A036353, A138281, A004189.

Sequence in context: A155629 A096596 * A146311 A212818 A195206 A081474

Adjacent sequences:  A001076 A001077 A001078 * A001080 A001081 A001082

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 24 22:11 EDT 2017. Contains 287008 sequences.