A254796 - OEIS

A254796

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

1, 4, 25, 200, 2025, 24300, 342225, 5475600, 98903025, 1978060500, 43616234025, 1046789616600, 27260146265625, 763284095437500, 22925783009390625, 733625056300500000, 24966177697226390625, 898782397100150062500, 34178697267502928765625

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OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..18.

T. J. Osler, The missing fractions in Brouncker’s sequence of continued fractions for Pi

FORMULA

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

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

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

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

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

EXAMPLE

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

MAPLE

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

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

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A182304 A060910 A195260 * A347585 A088159 A301363

Adjacent sequences: A254793 A254794 A254795 * A254797 A254798 A254799

KEYWORD

nonn,frac,easy

AUTHOR

Peter Bala, Feb 23 2015

STATUS

approved