A110260 Denominators in the coefficients that form the even-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.

1, 9, 225, 245, 99225, 480249, 1002001, 41409225, 2393453205, 4102737925, 940839860961, 4113258565689, 16802526820625, 246430431820125, 21147754404155625, 10036045423404225, 98363281194784809225

Offset: 1

Author

Paul D. Hanna, Jul 18 2005

Keywords

• frac
• nonn

Comments

Limit A110259(n)/a(n) = limit A110255(2*n)/A110256(2*n) = Pi.

Examples

			arctan(1/x) = 1/x - 1/(3*x^3) + 1/(5*x^5) - 1/(7*x^7) +-...
= [0; x, 3*x, 5/4*x, 28/9*x, 81/64*x, 704/225*x, 325/256*x,
768/245*x, 20825/16384*x, 311296/99225*x, 83349/65536*x,
1507328/480249*x, 1334025/1048576*x, 3145728/1002001*x,...]
= 1/(x + 1/(3*x + 1/(5/4*x + 1/(28/9*x + 1/(81/64*x +...))))).
The coefficients of x in the even-indexed partial quotients converge to Pi:
{3, 28/9, 704/225, 768/245, 311296/99225, ...}.
The coefficients of x in the odd-indexed partial quotients converge to 4/Pi:
{1, 5/4, 81/64, 325/256, 20825/16384, ...}.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A110259 (numerators), A110255/A110256 (continued fraction), A110257/A110258.

Programs

PARI

{a(n)=denominator(subst((contfrac( sum(k=0,2*n+2,(-1)^k/x^(2*k+1)/(2*k+1)),2*n+2))[2*n+1],x,1))}

Formula

a(n) = A110256(2*n).