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A087799 a(n) = 10*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 10, a(n) = (5+sqrt(24))^n + (5-sqrt(24))^n. 4
2, 10, 98, 970, 9602, 95050, 940898, 9313930, 92198402, 912670090, 9034502498, 89432354890, 885289046402, 8763458109130, 86749292044898, 858729462339850, 8500545331353602, 84146723851196170, 832966693180608098, 8245520207954884810 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

a(n+1)/a(n) converges to (5+sqrt(24)) = 9.8989794... a(0)/a(1)=2/10; a(1)/a(2)=10/98; a(2)/a(3)=98/970; a(3)/a(4)=970/9602; ... etc. Lim a(n)/a(n+1) as n approaches infinity = 0.10102051... = 1/(5+sqrt(24)) = (5-sqrt(24)).

Except for the first term, positive values of x (or y) satisfying x^2 - 10xy + y^2 + 96 = 0. - Colin Barker, Feb 25 2014

LINKS

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

P. Bala, Some simple continued fraction expansions for an infinite product, Part 1

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 (10,-1).

FORMULA

G.f.: (2-10*x)/(1-10*x+x^2). - Philippe Deléham, Nov 02 2008

From Peter Bala, Jan 06 2013: (Start)

Let F(x) = product {n = 0..inf} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 5 - sqrt(24). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.09989 80642 72052 68138 ... = 2 + 1/(10 + 1/(98 + 1/(970 + ...))).

Also F(-alpha) = 0.89989 78538 78393 34715 ... has the continued fraction representation 1 - 1/(10 - 1/(98 - 1/(970 - ...))) and the simple continued fraction expansion 1/(1 + 1/((10-2) + 1/(1 + 1/((98-2) + 1/(1 + 1/((970-2) + 1/(1 + ...))))))).

F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((10^2-4) + 1/(1 + 1/((98^2-4) + 1/(1 + 1/((970^2-4) + 1/(1 + ...))))))). Cf. A174503 and A005248.

(End)

a(-n) = a(n). - Michael Somos, Feb 25 2014

EXAMPLE

a(4) = 9602 = 10a(3) - a(2) = 10*970 - 98 =(5+sqrt(24))^4 + (5-sqrt(24))^4 =

9601.999895855 + 0.000104144 = 9602.

MATHEMATICA

a[0] = 2; a[1] = 10; a[n_] := 10a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 17}] (* Robert G. Wilson v, Jan 30 2004 *)

PROG

(Sage) [lucas_number2(n, 10, 1) for n in range(27)] # Zerinvary Lajos, Jun 25 2008

(PARI) polsym(x^2 - 10*x + 1, 20) \\ Charles R Greathouse IV, Jun 11 2011

(PARI) {a(n) = 2 * real( (5 + 2 * quadgen(24))^n )}; /* Michael Somos, Feb 25 2014 */

CROSSREFS

Cf. A086927, A036336. A005248, A174503.

Sequence in context: A193435 A132572 A069247 * A124214 A098279 A248615

Adjacent sequences:  A087796 A087797 A087798 * A087800 A087801 A087802

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

More terms from Colin Barker, Feb 25 2014

STATUS

approved

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Last modified November 18 14:12 EST 2017. Contains 294892 sequences.