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A159681 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 5*n(j)+1=a(j)*a(j) and 7*n(j)+1=b(j)*b(j) with positive integer numbers.

%I A159681 #18 Sep 27 2022 07:12:14
%S A159681 0,24,3432,487344,69199440,9825833160,1395199109304,198108447688032,
%T A159681 28130004372591264,3994262512460271480,567157146764985958920,
%U A159681 80532320578115545895184,11435022364945642531157232,1623692643501703123878431784,230552920354876897948206156120
%N A159681 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 5*n(j)+1=a(j)*a(j) and 7*n(j)+1=b(j)*b(j) with positive integer numbers.
%H A159681 Colin Barker, <a href="/A159681/b159681.txt">Table of n, a(n) for n = 1..450</a>
%H A159681 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (143,-143,1).
%F A159681 The a(j) recurrence is a(1)=1, a(2)=11, a(t+2) = 12*a(t+1) - a(t) resulting in terms 1, 11, 131, 1561, ... (A077417).
%F A159681 The b(j) recurrence is b(1)=1, b(2)=13, b(t+2) = 12*b(t+1) - b(t) resulting in terms 1, 13, 155, 1847, ... (A077416).
%F A159681 The n(j) recurrence is n(0)=n(1)=0, n(2)=24, n(t+3) = 143*(n(t+2) - n(t+1)) + n(t) resulting in terms 0, 0, 24, 3432, 487344, ... (this sequence).
%F A159681 G.f.: 24*x^2/((1-x)*(1-142*x+x^2)). - _R. J. Mathar_, Apr 20 2009
%F A159681 a(n) = (-12+(6+sqrt(35))*(71+12*sqrt(35))^(-n)-(-6+sqrt(35))*(71+12*sqrt(35))^n)/70. - _Colin Barker_, Jul 26 2016
%F A159681 a(n) = (6/35)*(ChebyshevU(n, 71) - 141*ChebyshevU(n-1, 71) - 1). - _G. C. Greubel_, Sep 27 2022
%p A159681 for a from 1 by 2 to 100000 do b:=sqrt((7*a*a-2)/5): if (trunc(b)=b) then
%p A159681 n:=(a*a-1)/5: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:
%p A159681 # Second program
%p A159681 seq((6/35)*(simplify(ChebyshevU(n,71) -141*ChebyshevU(n-1,71)) -1), n=1..30); # _G. C. Greubel_, Sep 27 2022
%t A159681 LinearRecurrence[{143,-143,1}, {0, 24, 3432}, 30] (* or *) CoefficientList[Series[24*x^2/((1-x)*(1-142*x+x^2)), {x,0,30}], x] (* _G. C. Greubel_, Jun 03 2018 *)
%o A159681 (PARI) concat(0, Vec(-24*x^2/((x-1)*(x^2-142*x+1)) + O(x^20))) \\ _Colin Barker_, Jul 26 2016
%o A159681 (PARI) a(n) = round((-12+(6+sqrt(35))*(71+12*sqrt(35))^(-n)-(-6+sqrt(35))*(71+12*sqrt(35))^n)/70) \\ _Colin Barker_, Jul 26 2016
%o A159681 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients( R!(24*x^2/((1-x)*(1-142*x+x^2)))); // _G. C. Greubel_, Jun 03 2018
%o A159681 (SageMath) [(6/35)*(-1 + chebyshev_U(n, 71) - 141*chebyshev_U(n-1, 71)) for n in range(1,30)] # _G. C. Greubel_, Sep 27 2022
%Y A159681 Cf. A077417, A077416, A157456.
%K A159681 nonn,easy
%O A159681 1,2
%A A159681 _Paul Weisenhorn_, Apr 19 2009
%E A159681 More terms from _R. J. Mathar_, Apr 20 2009