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A270371 Denominators of r-Egyptian fraction expansion for sqrt(1/2), where r = (1,1/4,1/9,1/16,...). 1
2, 2, 2, 3, 7, 7702, 1234163819, 1590823281229385753, 7255753768720849630767399215373753335, 44436679763085787755205863082559307822924182270889047678247210478391618529 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Suppose that r is a sequence of rational numbers r(k) <= 1 for k >= 1, and that x is an irrational number in (0,1). Let f(0) = x, n(k) = floor(r(k)/f(k-1)), and f(k) = f(k-1) - r(k)/n(k). Then x = r(1)/n(1)) + r(2)/(n(2) + r(3)/n(3) + ... , the r-Egyptian fraction for x.
See A269993 for a guide to related sequences.
LINKS
Eric Weisstein's World of Mathematics, Egyptian Fraction
EXAMPLE
sqrt(1/2) = 1/2 + 1/(4*2) + 1/(9*2) + 1/(16*3) + 1/(25*7) + ...
MATHEMATICA
r[k_] := 1/k^2; f[x_, 0] = x; z = 10;
n[x_, k_] := n[x, k] = Ceiling[r[k]/f[x, k - 1]]
f[x_, k_] := f[x, k] = f[x, k - 1] - r[k]/n[x, k]
x = Sqrt[1/2]; Table[n[x, k], {k, 1, z}]
PROG
(PARI) r(k) = 1/k^2;
f(k, x) = if (k==0, x, f(k-1, x) - r(k)/a(k, x); );
a(k, x=sqrt(1/2)) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016
CROSSREFS
Cf. A269993.
Sequence in context: A200918 A134890 A101360 * A351108 A184311 A110910
KEYWORD
nonn,frac,easy
AUTHOR
Clark Kimberling, Mar 20 2016
STATUS
approved

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Last modified March 2 12:34 EST 2024. Contains 370467 sequences. (Running on oeis4.)