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A159680 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 9*n(j) + 1 = a(j)*a(j) and 11*n(j) + 1 = b(j)*b(j) with positive integer numbers.

%I A159680 #15 Jul 07 2022 02:20:26
%S A159680 0,40,15960,6352080,2528111920,1006182192120,400457984351880,
%T A159680 159381271589856160,63433345634778399840,25246312181370213280200,
%U A159680 10047968814839710107119800,3999066341994023252420400240,1591618356144806414753212175760,633460106679290959048526025552280
%N A159680 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 9*n(j) + 1 = a(j)*a(j) and 11*n(j) + 1 = b(j)*b(j) with positive integer numbers.
%H A159680 Colin Barker, <a href="/A159680/b159680.txt">Table of n, a(n) for n = 1..350</a>
%H A159680 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (399,-399,1).
%F A159680 The a(j) recurrence is a(1)=1; a(2)=19; a(t+2) = 20*a(t+1) - a(t) resulting in terms 1, 19, 379, 7561, ... (A075839).
%F A159680 The b(j) recurrence is b(1)=1; b(2)=21; b(t+2) = 20*b(t+1) - b(t) resulting in terms 1, 21, 419, 8359, ... (A083043).
%F A159680 The n(j) recurrence is n(0)=n(1)=0; n(2)=40; n(t+3) = 399*(n(t+2) - n(t+1)) + n(t) resulting in terms 0, 0, 40, 15960, 6352080 as listed above
%F A159680 G.f.: 40*x^2/((1-x)*(1-398*x+x^2)). - _R. J. Mathar_, Apr 20 2009
%F A159680 a(n) = (-20 + (10 + 3*sqrt(11))*(199 + 60*sqrt(11))^(-n) + (10 - 3*sqrt(11))*(199 + 60*sqrt(11))^n)/198. - _Colin Barker_, Jul 26 2016
%F A159680 From _G. C. Greubel_, Jun 26 2022: (Start)
%F A159680 a(n) = (10/99)*( ChebyshevU(n, 199) - 397*ChebyshevU(n-1, 199) - 1 ).
%F A159680 E.g.f.: (10/99)*(exp(199*x)*( (3*sqrt(11)/10)*sinh(60*sqrt(11)*x) + cosh(60*sqrt(11)*x) ) - exp(x)). (End)
%p A159680 for a from 1 by 2 to 100000 do b:=sqrt((9*a*a-2)/7): if (trunc(b)=b) then
%p A159680 n:=(a*a-1)/7: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:
%t A159680 LinearRecurrence[{399,-399,1}, {0,40,15960}, 50] (* _G. C. Greubel_, Jun 03 2018 *)
%o A159680 (PARI) a(n) = round((-20+(10+3*sqrt(11))*(199+60*sqrt(11))^(-n)+(10-3*sqrt(11))*(199+60*sqrt(11))^n)/198) \\ _Colin Barker_, Jul 26 2016
%o A159680 (PARI) concat(0, Vec(-40*x^2/((x-1)*(x^2-398*x+1)) + O(x^20))) \\ _Colin Barker_, Jul 26 2016
%o A159680 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!(40*x^2/((1-x)*(1-398*x+x^2)))); // _G. C. Greubel_, Jun 03 2018
%o A159680 (SageMath) [(10/99)*(chebyshev_U(n, 199) -397*chebyshev_U(n-1, 199) -1) for n in (1..30)] # _G. C. Greubel_, Jun 26 2022
%Y A159680 Cf. A075839, A083043, A157456.
%K A159680 nonn,easy
%O A159680 1,2
%A A159680 _Paul Weisenhorn_, Apr 19 2009
%E A159680 More terms from _R. J. Mathar_, Apr 20 2009