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A087800 a(n) = 12a(n-1) - a(n-2), with a(0) = 2 and a(1) = 12. 3
2, 12, 142, 1692, 20162, 240252, 2862862, 34114092, 406506242, 4843960812, 57721023502, 687808321212, 8195978831042, 97663937651292, 1163771272984462, 13867591338162252, 165247324784962562, 1969100306081388492 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n+1)/a(n) converges to (6+sqrt(35)) = 11.9160797... a(0)/a(1)=2/12; a(1)/a(2)=12/142; a(2)/a(3)=142/1692; a(3)/a(4)=1692/20162; ... etc. Lim a(n)/a(n+1) as n approaches infinity = 0.0839202... = 1/(6+sqrt(35)) = (6-sqrt(35)).

Except for the first term, positive values of x (or y) satisfying x^2 - 12xy + y^2 + 140 = 0. - Colin Barker, Feb 25 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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 (12,-1).

FORMULA

a(n) = (6+sqrt(35))^n + (6-sqrt(35))^n.a(n) = 2*A023038(n).

G.f.: (2-12x)/(1-12x+x^2). - From Philippe Deléham, Nov 17 2008

a(-n) = a(n). - Michael Somos, May 28 2014

EXAMPLE

a(4) = 20162 = 12a(3) - a(2) = 12*1692 - 142 = (6+sqrt(35))^4 + (6-sqrt(35))^4 =

20161.9999504 + 0.00004959 = 20162.

G.f. = 2 + 12*x + 142*x^2 + 1692*x^3 + 20162*x^4 + 240252*x^5 + ...

MATHEMATICA

a[0] = 2; a[1] = 12; a[n_] := 12a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 17}] (* Robert G. Wilson v, Jan 30 2004 *)

CoefficientList[Series[(2 - 12 x)/(1 - 12 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 28 2014 *)

a[ n_] := 2 ChebyshevT[ n, 6]; (* Michael Somos, May 28 2014 *)

PROG

(Sage) [lucas_number2(n, 12, 1) for n in xrange(1, 20)] # Zerinvary Lajos, Jun 25 2008

(PARI) Vec((2-12*x)/(1-12*x+x^2) + O(x^100)) \\ Colin Barker, Feb 25 2014

(PARI) {a(n) = 2 * polchebyshev( n, 1, 6)}; /* Michael Somos, May 28 2014 */

CROSSREFS

Cf. A009747, A086928, A001927, A023038.

Sequence in context: A091144 A275829 A240387 * A009747 A208866 A067601

Adjacent sequences:  A087797 A087798 A087799 * A087801 A087802 A087803

KEYWORD

easy,nonn

AUTHOR

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Oct 11 2003

STATUS

approved

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Last modified August 24 05:48 EDT 2017. Contains 291052 sequences.