A254796 - OEIS

%I #25 Sep 08 2022 08:46:11

%S 1,4,25,200,2025,24300,342225,5475600,98903025,1978060500,43616234025,

%T 1046789616600,27260146265625,763284095437500,22925783009390625,

%U 733625056300500000,24966177697226390625,898782397100150062500,34178697267502928765625

%N Denominators of the convergents of the generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))).

%C The generalized continued fraction 2 + 1^2/(4 + 3^2/(4 + 5^2/(4 + ... ))) represents the constant L^2/Pi = 2.188439... = A254794, where L is the lemniscate constant A062539. See A254795 for the numerators of the convergents of the continued fraction.

%H T. J. Osler, <a href="http://www.rowan.edu/colleges/csm/departments/math/facultystaff/osler/130%20Missing%20fractions%20in%20Brouncker.pdf">The missing fractions in Brouncker’s sequence of continued fractions for Pi</a>

%F a(2*n) = A007696(n+1)^2 = ( Product {k = 0..n} 4*k + 1 )^2.

%F a(2*n-1) = 4*n*A007696(n)^2 = 4*n * ( Product {k = 0..n-1} 4*k + 1 )^2.

%F a(n) = 4*a(n-1) + (2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 4.

%F a(2*n+1) = 4*(n + 1)*a(2*n); a(2*n) = (4*n + 2)*a(2*n-1) + a(2*n-2).

%F Empirical e.g.f.: ((-Q(1/2, -3)-Q(-1/2, -3))*P(1/2, (2*x+3)/(2*x-1))+Q(1/2, (2*x+3)/(2*x-1))*(P(1/2, -3)+P(-1/2, -3)))/((1-2*x)^(3/2)*(-Q(-1/2, -3)*P(1/2, -3)+Q(1/2, -3)*P(-1/2, -3))) where P and Q are Legendre functions of the first and second kinds. - _Robert Israel_, Feb 24 2015

%e 54/25 = 2.16, 441/200 = 2.205 etc approach 2.188..

%p a[0] := 1: a[1] := 4:

%p for n from 2 to 18 do a[n] := 4*a[n-1] + (2*n-1)^2*a[n-2] end do:

%p seq(a[n], n = 0 .. 18);

%t RecurrenceTable[{a[0] == 1, a[1] == 4, a[n] == 4 a[n - 1] + (2 n - 1)^2 a[n - 2]}, a, {n, 20}] (* _Vincenzo Librandi_, Feb 24 2015 *)

%o (Magma) I:=[1,4]; [n le 2 select I[n] else 4*Self(n-1)+(2*n-3)^2*Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Feb 24 2015

%Y Cf. A007696, A024199, A062539, A254794, A254795.

%K nonn,frac,easy

%O 0,2

%A _Peter Bala_, Feb 23 2015