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A067902 a(n) = 14*a(n-1) - a(n-2); a(0) = 2, a(1) = 14. 3
2, 14, 194, 2702, 37634, 524174, 7300802, 101687054, 1416317954, 19726764302, 274758382274, 3826890587534, 53301709843202, 742397047217294, 10340256951198914, 144021200269567502, 2005956546822746114, 27939370455248878094, 389145229826661547202, 5420093847118012782734 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Solves for x in x^2 - 3*y^2 = 4.

For n>0, a(n)+2 is the number of dimer tilings of a 4 X 2n Klein bottle (cf. A103999).

LINKS

Table of n, a(n) for n=0..19.

Tanya Khovanova, Recursive Sequences

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for linear recurrences with constant coefficients, signature (14,-1).

FORMULA

G.f.: 2*(1-7*x)/(1-14*x+x^2). - N. J. A. Sloane, Nov 22 2006

a(n) = p^n + q^n, where p = 7 + 4*sqrt(3) and q = 7 - 4*sqrt(3). - Tanya Khovanova, Feb 06 2007

a(n) = 2*A011943(n+1). - R. J. Mathar, Sep 27 2014

EXAMPLE

G.f. = 2 + 14*x + 194*x^2 + 2702*x^3 + 37634*x^4 + 524174*x^5 + ...

MAPLE

a := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(14) fi: 14*a(n-1)-a(n-2): end: for n from 0 to 30 do printf(`%d, `, a(n)) od:

MATHEMATICA

a[0] = 2; a[1] = 14; a[n_] := 14a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Jan 30 2004 *)

PROG

(Sage) [lucas_number2(n, 14, 1) for n in xrange(0, 20)] # Zerinvary Lajos, Jun 26 2008

(MAGMA) [Floor((2+Sqrt(3))^(2*n)+(1+Sqrt(3))^(-n)): n in [0..19]]; // Vincenzo Librandi, Mar 31 2011

CROSSREFS

Cf. A067900.

Row 2 * 2 of array A188644

Sequence in context: A074655 A268005 A158102 * A132611 A156327 A047796

Adjacent sequences:  A067899 A067900 A067901 * A067903 A067904 A067905

KEYWORD

nonn,easy

AUTHOR

Lekraj Beedassy, May 13 2003

STATUS

approved

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Last modified June 22 08:00 EDT 2017. Contains 288605 sequences.