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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.

%I A159664 #30 Jun 12 2025 11:44:32
%S A159664 1,23,551,13201,316273,7577351,181540151,4349386273,104203730401,
%T A159664 2496540143351,59812759710023,1433009692897201,34332419869822801,
%U A159664 822545067182850023,19706749192518577751,472139435553263016001,11311639704085793806273,271007213462505788334551
%N 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.
%C A159664 Positive values of x (or y) satisfying x^2 - 24*x*y + y^2 + 22 = 0. - _Colin Barker_, Feb 19 2014
%H A159664 Vincenzo Librandi, <a href="/A159664/b159664.txt">Table of n, a(n) for n = 1..200</a>
%H A159664 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (24,-1).
%F A159664 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, ... (this sequence).
%F A159664 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, ... (A159661).
%F A159664 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, ... (A159665).
%F A159664 G.f.: x*(1-x)/(1 - 24*x + x^2). - _Colin Barker_, Feb 19 2014
%F A159664 a(n) = (12+sqrt(143))^(-n)*(13+sqrt(143)-(-13+sqrt(143))*(12+sqrt(143))^(2*n))/26. - _Colin Barker_, Jul 25 2016
%F A159664 a(n) = A077423(n-1) - A077423(n-2). - _G. C. Greubel_, Sep 27 2022
%p A159664 for a from 1 by 2 to 100000 do b:=sqrt((13*a*a-2)/11): if (trunc(b)=b) then
%p A159664 n:=(a*a-1)/11: La:=[La),a]:Lb:=[op(Lb),b]: Ln:=[op(Ln),n]: end if: end do:
%p A159664 # Second program
%p A159664 seq(simplify(ChebyshevU(n-1,12) - ChebyshevU(n-2,12)), n=1..30); # _G. C. Greubel_, Sep 27 2022
%t A159664 CoefficientList[Series[(1-x)/(1-24x+x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 21 2014 *)
%t A159664 LinearRecurrence[{24,-1}, {1,23}, 30] (* _G. C. Greubel_, Sep 27 2022 *)
%o A159664 (PARI) Vec(x*(1-x)/(1-24*x+x^2) + O(x^100)) \\ _Colin Barker_, Feb 19 2014
%o A159664 (Magma) [n le 2 select 23^(n-1) else 24*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 21 2014
%o A159664 (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
%o A159664 (SageMath)
%o A159664 def A159664(n): return chebyshev_U(n-1,12) - chebyshev_U(n-2,12)
%o A159664 [A159664(n) for n in range(1,30)] # _G. C. Greubel_, Sep 27 2022
%Y A159664 Cf. A077423, A157456, A159661, A159665.
%Y A159664 Cf. similar sequences listed in A238379.
%K A159664 nonn,easy
%O A159664 1,2
%A A159664 _Paul Weisenhorn_, Apr 19 2009
%E A159664 More terms from _Colin Barker_, Feb 19 2014