cp's OEIS Frontend

A159683 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 3*n(j) + 1 = a(j)*a(j) and 5*n(j) + 1 = b(j)*b(j) with positive integer numbers.

%I A159683 #26 Sep 28 2022 18:53:23
%S A159683 0,16,1008,62496,3873760,240110640,14882985936,922505017408,
%T A159683 57180428093376,3544264036771920,219687189851765680,
%U A159683 13617061506772700256,844038126230055650208,52316746764756677612656,3242794261288683956334480,201000927453133648615125120
%N A159683 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 3*n(j) + 1 = a(j)*a(j) and 5*n(j) + 1 = b(j)*b(j) with positive integer numbers.
%H A159683 Colin Barker, <a href="/A159683/b159683.txt">Table of n, a(n) for n = 1..559</a>
%H A159683 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (63,-63,1).
%F A159683 The a(j) recurrence is a(1)=1, a(2)=7, a(t+2) = 8*a(t+1) - a(t) resulting in terms 1, 7, 55. 433, 3409, ... (A070997).
%F A159683 The b(j) recurrence is b(1)=1, b(2)=9, b(t+2) = 8*b(t+1) - b(t) resulting in terms 1, 9, 71, 559, 4401, ... (A057080).
%F A159683 The n(j) recurrence is n(0) = n(1) = 0, n(2)=16, n(t+3) = 63*(n(t+2) - n(t+1)) + n(t) resulting in terms 0, 0, 16, 1008, 62496, ... (this sequence).
%F A159683 From _Colin Barker_, Sep 25 2015: (Start)
%F A159683 a(n) = 63*a(n-1) - 63*a(n-2) + a(n-3) for n>3.
%F A159683 G.f.: 16*x^2 / ((1-x)*(1-62*x+x^2)). (End)
%F A159683 a(n) = (-8+(4+sqrt(15))*(31+8*sqrt(15))^(-n) -(-4+sqrt(15))*(31+8*sqrt(15))^n)/30. - _Colin Barker_, Mar 03 2016
%F A159683 a(n) = (4/15)*(-1 + ChebyshevU(n, 31) - 61*ChebyshevU(n-1, 31)). - _G. C. Greubel_, Sep 27 2022
%p A159683 for a from 1 by 2 to 100000 do b:=sqrt((5*a*a-2)/3): if (trunc(b)=b) then
%p A159683 n:=(a*a-1)/3: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:
%p A159683 # Second program
%p A159683 seq((4/15)*(simplify(ChebyshevU(n, 31) - 61*ChebyshevU(n-1, 31)) -1), n=1..30); # _G. C. Greubel_, Sep 27 2022
%t A159683 CoefficientList[Series[16*x/((1-x)*(1-62*x+x^2)), {x, 0, 30}], x] (* _G. C. Greubel_, Jun 02 2018 *)
%t A159683 LinearRecurrence[{63,-63,1},{0,16,1008},30] (* _Harvey P. Dale_, May 07 2022 *)
%o A159683 (PARI) concat(0, Vec(16*x^2/((1-x)*(1-62*x+x^2)) + O(x^30))) \\ _Colin Barker_, Sep 25 2015
%o A159683 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!(16*x^2/((1-x)*(1-62*x+x^2)))); // _G. C. Greubel_, Jun 02 2018
%o A159683 (SageMath) [(4/15)*(-1 + chebyshev_U(n, 31) - 61*chebyshev_U(n-1, 31)) for n in range(1,30)] # _G. C. Greubel_, Sep 27 2022
%Y A159683 Cf. A057080, A070997, A157456, A245031.
%K A159683 nonn,easy
%O A159683 1,2
%A A159683 _Paul Weisenhorn_, Apr 19 2009