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A159661 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 elements. the solutions of the 2 equations problem: 11*n(j)+1=a(j)*a(j); 13*n(j)+1=b(j)*b(j); with integer numbers. 1
1, 25, 599, 14351, 343825, 8237449, 197354951, 4728281375, 113281398049, 2714025271801, 65023325125175, 1557845777732399, 37323275340452401, 894200762393125225, 21423495022094552999, 513269679767876146751, 12297048819406932969025, 294615901985998515109849 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Colin Barker, Table of n, a(n) for n = 1..725

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 a(j) terms 1,23,551,13201,316273,7577351,181540151,4349386273.

The b(j) recurrence is b(1)=1; b(2)=23; b(t+2)=24*b(t+1)-b(t); resulting in b(j) terms 1,25,599,14351,343825,8237449 as listed above.

The n(j) recurrence is n(0)=n(1)=0; n(2)=48; n(t+3)=575*(n(t+2)-n(t+1))+n(t) resulting in n(j) terms 0,0,48,27600,15842400,9093510048,5219658925200.

From Colin Barker, Sep 25 2015: (Start)

a(n) = 24*a(n-1)-a(n-2) for n>2.

G.f.: x*(x+1) / (x^2-24*x+1).

(End)

a(n) = (12+sqrt(143))^(-n)*(-11-sqrt(143)+(-11+sqrt(143))*(12+sqrt(143))^(2*n))/22. - Colin Barker, Jul 26 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^2-1)/C: La:=[op(La), a]: Lb:=[op(Lb), b]: Ln:=[op(Ln), n]: endif: enddo:

PROG

(PARI) Vec(x*(x+1)/(x^2-24*x+1) + O(x^20)) \\ Colin Barker, Sep 25 2015

(PARI) a(n) = round((12+sqrt(143))^(-n)*(-11-sqrt(143)+(-11+sqrt(143))*(12+sqrt(143))^(2*n))/22) \\ Colin Barker, Jul 26 2016

CROSSREFS

Cf. A157456.

Sequence in context: A239822 A264220 A228827 * A104643 A162811 A163175

Adjacent sequences:  A159658 A159659 A159660 * A159662 A159663 A159664

KEYWORD

nonn,easy

AUTHOR

Paul Weisenhorn, Apr 19 2009

STATUS

approved

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Last modified October 16 20:55 EDT 2019. Contains 328103 sequences. (Running on oeis4.)