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A001091 a(n) = 8a(n-1) - a(n-2); a(0) = 1, a(1) = 4.
(Formerly M3637 N1479)
18
1, 4, 31, 244, 1921, 15124, 119071, 937444, 7380481, 58106404, 457470751, 3601659604, 28355806081, 223244789044, 1757602506271, 13837575261124, 108942999582721, 857706421400644, 6752708371622431, 53163960551578804 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(15+30k)-1 and a(15+30k)+1 are consecutive odd powerful numbers. The first pair is 13837575261124+-1. See A076445. - T. D. Noe, May 04 2006

Numbers n such that 15*(n^2-1) is a square. - Vincenzo Librandi, Jul 08 2010

This sequence gives the values of x in solutions of the Diophantine equation x^2 - 15*y^2 = 1. The corresponding y values are in A001090. - Vincenzo Librandi, Nov 12 2010 [edited by Jon E. Schoenfield, May 04 2014]

The square root of 15*(n^2-1) at those numbers = 5 * A136325. - Richard R. Forberg, Nov 22 2013

For the above Diophantine equation x^2-15*y^2=1, x + y = A105426. - Richard R. Forberg, Nov 22 2013

a(n) solves for x in the Diophantine equation floor(3*x^2/5)= y^2. The corresponding y solutions are provided by A136325(n).  x + y = A070997(n). - Richard R. Forberg, Nov 22 2013

Except for the first term, values of x (or y) in the solutions to x^2 - 8xy + y^2 + 15 = 0. - Colin Barker, Feb 05 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

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..200

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

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 sequences related to Chebyshev polynomials.

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

FORMULA

For all elements x of the sequence, 15*x^2 -15 is a square. Lim. n -> Inf. a(n)/a(n-1) = 4 + sqrt(15). - Gregory V. Richardson, Oct 11 2002

a(n) = ((4+sqrt(15))^n + (4-sqrt(15))^n)/2.

a(n) = 4*S(n-1, 8)-S(n-2, 8) = (S(n, 8)-S(n-2, 8))/2, n>=1; S(n, x) := U(n, x/2) with Chebyshev's polynomials of the 2nd kind, A049310, with S(-1, x) := 0 and S(-2, x) := -1.

a(n) = T(n, 4) with Chebyshev's polynomials of the first kind; see A053120.

G.f.: (1-4*x)/(1-8*x+x^2). a(n)=a(-n). - Ralf Stephan, Jun 06 2005

a(n)a(n+3) - a(n+1)a(n+2) = 120. - Ralf Stephan, Jun 06 2005

MAPLE

A001091:=-(-1+4*z)/(1-8*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

MATHEMATICA

LinearRecurrence[{8, -1}, {1, 4}, 20] (* Harvey P. Dale, May 01 2014 *)

PROG

(PARI) a(n)=subst(poltchebi(n), x, 4)

(PARI) a(n)=n=abs(n); polcoeff((1-4*x)/(1-8*x+x^2)+x*O(x^n), n) /* Michael Somos, Jun 07 2005 */

CROSSREFS

a(n) = sqrt{15*[(A001090(n))^2]+1}.

Sequence in context: A136284 A183911 A039765 * A077615 A039306 A265949

Adjacent sequences:  A001088 A001089 A001090 * A001092 A001093 A001094

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Aug 25 2000

Chebyshev comments from Wolfdieter Lang, Oct 31 2002

STATUS

approved

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Last modified April 25 23:17 EDT 2017. Contains 285426 sequences.