|
|
A110257
|
|
Numerators in the coefficients that form the odd-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.
|
|
7
|
|
|
1, 5, 81, 325, 20825, 83349, 1334025, 5337189, 1366504425, 5466528925, 87470372561, 349899121845, 22394407746529, 89580335298125, 1433319858545625, 5733391194015525, 5871086572691471625
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
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, ...}.
|
|
MAPLE
|
a := n -> (4*n+1)*binomial(2*n, n)^2/4^(add(i, i=convert(n, base, 2)));
|
|
MATHEMATICA
|
a[n_] := (4n+1) Binomial[2n, n]^2 / 4^DigitCount[n, 2, 1];
|
|
PROG
|
(PARI) {a(n)=numerator(subst((contfrac( sum(k=0, 2*n+1, (-1)^k/x^(2*k+1)/(2*k+1)), 2*n+2))[2*n], x, 1))}
|
|
CROSSREFS
|
|
|
KEYWORD
|
frac,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|