OFFSET
0,2
COMMENTS
See the numerator sequence A224445. The rationals r(n) are the partial sums of the series 2*sqrt(1 - 1/4), representing sqrt(3).
Looks as if a(n) are all powers of 2, a(n) = 2^b(n) with b(n) = 0, 2, 6, 9, 14, 17, 21, 24, 30, 33, 37, 40, 45, 48, ... - Peter Luschny, Apr 05 2014
The sequence appears in Newton's De Analysi (1669) and De Methodis (1671) in connection to the literal equation y^3 + (a^2)*y - 2*a^3 + a*x*y - x^3 = 0. y is expressed as an infinite series of x. a(n) appears in the denominators of the infinite series. For more details see the links to Charles Hutton's book and Naoki Osada's paper. - Raul Prisacariu, Mar 22 2026
REFERENCES
H. K. Strick, Geschichten aus der Mathematik, Spektrum Spezial 2/2009, p. 45 (on Newton).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Charles Hutton, APPROXIMATION, A mathematical and philosophical dictionary Vol. 1 (1795), 131-132.
Naoki Osada, Literal resolution of affected equations by Isaac Newton, RIMS Kôkyûroku Bessatsu B73 (2019), 1-20.
FORMULA
a(n) = denominator(r(n)), n >= 0, with the rationals (in lowest terms) r(n) = 2*(1 - 2*Sum_{k=1..n} C(k-1)/2^(4*k)), with the Catalan numbers C(n) = A000108(n).
EXAMPLE
a(2) = 64 because r(2) = 111/64 = A224445(2)/a(2).
MAPLE
A224446 := proc(n) (x/(exp(x)-1))^(3/2)*exp(x/2);
-pochhammer(1/2, n-1)*coeff(series(%, x, n+2), x, n); denom(%) end:
seq(A224446(n), n=0..17); # Peter Luschny, Apr 05 2014
MATHEMATICA
r[n_] := 2*(1 - 2*Sum[ CatalanNumber[k - 1]/4^(2*k), {k, 1, n}]); Table[r[n], {n, 0, 17}] // Denominator (* Jean-François Alcover, Apr 09 2013 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Wolfdieter Lang, Apr 09 2013
STATUS
approved