A040012 -id:A040012 - cp's OEIS frontend

A040930 Continued fraction for sqrt(962).

31, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62

Offset: 0

N. J. A. Sloane

			31 + 1/(62 + 1/(62 + 1/(62 + 1/(62 + ...)))) = sqrt(962).

Cf. A042860/A042861 (convergents).

Continued fraction for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040000 (contfrac(sqrt(2)) = (1,2,2,...)), A040002, A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870 (contfrac(sqrt(901)) = (30,60,60,...)).

with(numtheory): Digits := 300: convert(evalf(sqrt(962)),confrac);

PadRight[{31},100,62] (* Harvey P. Dale, Sep 18 2012 *)

G.f.: 31*(1+x)/(1-x). - Colin Barker, Aug 11 2012
From Elmo R. Oliveira, Feb 16 2024: (Start)
a(n) = 62 for n >= 1.
E.g.f.: 62*exp(x) - 31.
a(n) = 31*A040000(n). (End)

A121740 Solutions to the Pell equation x^2 - 17y^2 = 1 (y values).

Original entry on oeis.org

0, 8, 528, 34840, 2298912, 151693352, 10009462320, 660472819768, 43581196642368, 2875698505576520, 189752520171407952, 12520790632807348312, 826182429245113580640, 54515519539544688973928

Offset: 1

Rick L. Shepherd, Jul 31 2006

After initial term this sequence bisects A041025. See A099370 for corresponding x values. a(n+1)/a(n) apparently converges to (4+sqrt(17))^2.

The first solution to the equation x^2 - 17*y^2 = 1 is (X(1); Y(1)) = (1, 0) and the other solutions are defined by: (X(n), Y(n))= (33*X(n-1) + 136*Y(n-1), 8*X(n-1) + 33*Y(n-1)) with n >= 2. - Mohamed Bouhamida, Jan 16 2020

			A099370(1)^2 - 17*a(1)^2 = 33^2 - 17*8^2 = 1089 - 1088 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Tanya Khovanova, Recursive Sequences.
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for linear recurrences with constant coefficients, signature (66,-1).

Cf. A099370, A041025, A040012.

I:=[0, 8]; [n le 2 select I[n] else 66*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 18 2011

LinearRecurrence[{66,-1},{0,8},30] (* Vincenzo Librandi, Dec 18 2011 *)

makelist(expand(((33+8*sqrt(17))^n - (33-8*sqrt(17))^n) /(4*sqrt(17)/2)), n, 0, 16); /* Vincenzo Librandi, Dec 18 2011 */

\\ Program uses fact that continued fraction for sqrt(17) = [4,8,8,...].
print1("0, "); forstep(n=2,40,2,v=vector(n,i,if(i>1,8,4)); print1(contfracpnqn(v)[2,1],", "))

a(n) = ((33+8*sqrt(17))^(n-1) - (33-8*sqrt(17))^(n-1))/(2*sqrt(17)).
From Mohamed Bouhamida, Feb 07 2007: (Start)
a(n) = 65*(a(n-1) + a(n-2)) - a(n-3).
a(n) = 67*(a(n-1) - a(n-2)) + a(n-3). (End)
From Philippe Deléham, Nov 18 2008: (Start)
a(n) = 66*a(n-1) - a(n-2) for n > 1; a(1)=0, a(2)=8.
G.f.: 8*x^2/(1 - 66*x + x^2). (End)
E.g.f.: (1/17)*exp(33*x)*(33*sqrt(17)*sinh(8*sqrt(17)*x) + 136*(1 - cosh(8*sqrt(17)*x))). - Stefano Spezia, Feb 08 2020

Offset changed from 0 to 1 and g.f. adapted by Vincenzo Librandi, Dec 18 2011

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A040930 Continued fraction for sqrt(962).

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A121740 Solutions to the Pell equation x^2 - 17y^2 = 1 (y values).

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