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A159664 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 11*n(j)+1=a(j)*a(j) and 13*n(j)+1=b(j)*b(j); with positive integer numbers. 3
1, 23, 551, 13201, 316273, 7577351, 181540151, 4349386273, 104203730401, 2496540143351, 59812759710023, 1433009692897201, 34332419869822801, 822545067182850023, 19706749192518577751, 472139435553263016001, 11311639704085793806273, 271007213462505788334551 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Positive values of x (or y) satisfying x^2 - 24xy + y^2 + 22 = 0. - Colin Barker, Feb 19 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (24,-1).

FORMULA

The a(j) recurrence is a(1)=1; a(2)=23; a(t+2)=24*a(t+1)-a(t) resulting in terms 1, 23, 551, 13201 as listed above.

The b(j) recurrence is b(1)=1; b(2)=25; b(t+2)=24*b(t+1)-b(t) resulting in terms 1, 25, 599, 14351.

The n(j) recurrence is n(0)=n(1)=1; n(2)=48; n(t+3)=575*(n(t+2)-n(t+1))+n(t) resulting in terms 0, 0, 48, 27600, 15842400.

G.f.: -x*(x-1) / (x^2-24*x+1). - Colin Barker, Feb 19 2014

a(n) = (12+sqrt(143))^(-n)*(13+sqrt(143)-(-13+sqrt(143))*(12+sqrt(143))^(2*n))/26. - Colin Barker, Jul 25 2016

MAPLE

for a from 1 by 2 to 100000 do b:=sqrt((13*a*a-2)/11): if (trunc(b)=b) then

n:=(a*a-1)/11: La:=[La), a]:Lb:=[op(Lb), b]: Ln:=[op(Ln), n]: end if: end do:

MATHEMATICA

CoefficientList[Series[(1 - x)/(x^2 - 24 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 21 2014 *)

PROG

(PARI) Vec(-x*(x-1)/(x^2-24*x+1) + O(x^100)) \\ Colin Barker, Feb 19 2014

(MAGMA) I:=[1, 23]; [n le 2 select I[n] else 24*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 21 2014

(PARI) a(n) = round((12+sqrt(143))^(-n)*(13+sqrt(143)-(-13+sqrt(143))*(12+sqrt(143))^(2*n))/26) \\ Colin Barker, Jul 25 2016

CROSSREFS

Cf. A157456.

Cf. similar sequences listed in A238379.

Sequence in context: A136670 A062360 A062511 * A158631 A196536 A231261

Adjacent sequences:  A159661 A159662 A159663 * A159665 A159666 A159667

KEYWORD

nonn,easy

AUTHOR

Paul Weisenhorn, Apr 19 2009

EXTENSIONS

More terms from Colin Barker, Feb 19 2014

STATUS

approved

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Last modified February 23 04:57 EST 2018. Contains 299473 sequences. (Running on oeis4.)