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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 (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

Clark Kimberling, Table of n, a(n) for n = 1..10

Eric Weisstein's World of Mathematics, Egyptian Fraction

Index entries for sequences related to Egyptian fractions

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

Adjacent sequences:  A270374 A270375 A270376 * A270378 A270379 A270380

KEYWORD

nonn,frac,easy

AUTHOR

Clark Kimberling, Mar 20 2016

STATUS

approved

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Last modified July 14 10:35 EDT 2020. Contains 335722 sequences. (Running on oeis4.)