A270400 Denominators of r-Egyptian fraction expansion for Pi - 3, where r(k) = 1/Fibonacci(k+1).

8, 31, 719, 13821369, 220130496618675, 172884943957152518892033300539, 41439757913276192740376226557217884637591923384027133159856, 1439356540409399527843567497648639858961697820320457801939412357033200260469148238882098706866046526043076854124526724

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

Pi - 3 = 1/8 + 1/(2*31) + 1/(3*719) + ...

Mathematica

r[k_] := 1/Fibonacci[k+1]; 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/fibonacci(k+1);
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 22 2016

Crossrefs

Cf. A269993, A000045, A000796.

Sequence in context: A270523 A332916 A270353 * A269999 A298944 A230309

Adjacent sequences: A270397 A270398 A270399 * A270401 A270402 A270403

Keyword

nonn,frac,easy

Author

Clark Kimberling, Mar 22 2016

Status

approved