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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.

%I A159661 #17 Dec 26 2023 12:23:39
%S A159661 1,25,599,14351,343825,8237449,197354951,4728281375,113281398049,
%T A159661 2714025271801,65023325125175,1557845777732399,37323275340452401,
%U A159661 894200762393125225,21423495022094552999,513269679767876146751,12297048819406932969025,294615901985998515109849
%N 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.
%H A159661 Colin Barker, <a href="/A159661/b159661.txt">Table of n, a(n) for n = 1..725</a>
%H A159661 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (24,-1).
%F A159661 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.
%F A159661 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.
%F A159661 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.
%F A159661 From _Colin Barker_, Sep 25 2015: (Start)
%F A159661 a(n) = 24*a(n-1)-a(n-2) for n>2.
%F A159661 G.f.: x*(1+x) / (1 - 24*x + x^2). (End)
%F A159661 a(n) = (12+sqrt(143))^(-n)*(-11 - sqrt(143) + (-11+sqrt(143))*(12+sqrt(143))^(2*n))/22. - _Colin Barker_, Jul 26 2016
%F A159661 From _G. C. Greubel_, Jun 25 2022: (Start)
%F A159661 a(n) = ChebyshevU(n-1, 12) + Chebyshev(n-2, 12).
%F A159661 E.g.f.: exp(12*x)*(cosh(sqrt(143)*x) + sqrt(13/11)*sinh(sqrt(143)*x)). (End)
%p A159661 for a from 1 by 2 to 100000 do b:=sqrt((13*a*a-2)/11): if (trunc(b)=b) then
%p A159661 n:=(a^2-1)/C: La:=[op(La),a]: Lb:=[op(Lb),b]: Ln:=[op(Ln),n]: endif: enddo:
%t A159661 LinearRecurrence[{24,-1}, {1,25}, 31] (* _G. C. Greubel_, Jun 25 2022 *)
%o A159661 (PARI) Vec(x*(x+1)/(x^2-24*x+1) + O(x^20)) \\ _Colin Barker_, Sep 25 2015
%o A159661 (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
%o A159661 (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
%o A159661 (SageMath) [chebyshev_U(n-1, 12) + chebyshev_U(n-2, 12) for n in (1..30)] # _G. C. Greubel_, Jun 25 2022
%Y A159661 Cf. A077423, A157456.
%K A159661 nonn,easy
%O A159661 1,2
%A A159661 _Paul Weisenhorn_, Apr 19 2009