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, 25, 599, 14351, 343825, 8237449, 197354951, 4728281375, 113281398049, 2714025271801, 65023325125175, 1557845777732399, 37323275340452401, 894200762393125225, 21423495022094552999, 513269679767876146751, 12297048819406932969025, 294615901985998515109849

Offset: 1

Author

Paul Weisenhorn, Apr 19 2009

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..725
• Index entries for linear recurrences with constant coefficients, signature (24,-1).

Crossrefs

Cf. A077423, A157456.

Programs

Magma

[n le 2 select 24*n-23 else 24*Self(n-1) -Self(n-2): n in [1..31]]; // G. C. Greubel, Jun 25 2022

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:

Mathematica

LinearRecurrence[{24,-1}, {1,25}, 31] (* G. C. Greubel, Jun 25 2022 *)

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

SageMath

[chebyshev_U(n-1, 12) + chebyshev_U(n-2, 12) for n in (1..30)] # G. C. Greubel, Jun 25 2022

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*(1+x) / (1 - 24*x + x^2). (End)

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

From G. C. Greubel, Jun 25 2022: (Start)

a(n) = ChebyshevU(n-1, 12) + Chebyshev(n-2, 12).

E.g.f.: exp(12*x)*(cosh(sqrt(143)*x) + sqrt(13/11)*sinh(sqrt(143)*x)). (End)