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A270377
Denominators of r-Egyptian fraction expansion for Pi - 3, where r = (1,1/4,1/9,1/16,...).
1
8, 16, 115, 42517, 2725016283, 22037592325978294230, 376949052509622237440534036730873293477, 162105898616252691011784334305248213903014362390225130418238883927812046205359
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.
EXAMPLE
1/Pi = 1/8 + 1/(4*16) + 1/(9*115) + 1/(16*42517) + ...
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 = Pi - 3; 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=Pi-3) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 21 2016
CROSSREFS
Cf. A269993.
Sequence in context: A083086 A230930 A080452 * A264472 A264478 A277364
KEYWORD
nonn,frac,easy,changed
AUTHOR
Clark Kimberling, Mar 20 2016
STATUS
approved