A159681 - OEIS

%I #18 Sep 27 2022 07:12:14

%S 0,24,3432,487344,69199440,9825833160,1395199109304,198108447688032,

%T 28130004372591264,3994262512460271480,567157146764985958920,

%U 80532320578115545895184,11435022364945642531157232,1623692643501703123878431784,230552920354876897948206156120

%N 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 Colin Barker, <a href="/A159681/b159681.txt">Table of n, a(n) for n = 1..450</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (143,-143,1).

%F 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 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 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 G.f.: 24*x^2/((1-x)*(1-142*x+x^2)). - _R. J. Mathar_, Apr 20 2009

%F 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 a(n) = (6/35)*(ChebyshevU(n, 71) - 141*ChebyshevU(n-1, 71) - 1). - _G. C. Greubel_, Sep 27 2022

%p for a from 1 by 2 to 100000 do b:=sqrt((7*a*a-2)/5): if (trunc(b)=b) then

%p n:=(a*a-1)/5: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:

%p # Second program

%p seq((6/35)*(simplify(ChebyshevU(n,71) -141*ChebyshevU(n-1,71)) -1), n=1..30); # _G. C. Greubel_, Sep 27 2022

%t 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 (PARI) concat(0, Vec(-24*x^2/((x-1)*(x^2-142*x+1)) + O(x^20))) \\ _Colin Barker_, Jul 26 2016

%o (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 (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 (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 Cf. A077417, A077416, A157456.

%K nonn,easy

%O 1,2

%A _Paul Weisenhorn_, Apr 19 2009

%E More terms from _R. J. Mathar_, Apr 20 2009