A270396 Denominators of r-Egyptian fraction expansion for sqrt(2) - 1, where r(k) = 1/Fibonacci(k+1).

3, 7, 36, 1040, 1378784, 2783678150237, 20812561896916543523976387, 398006071848302987834283599453836703483929049938762, 105246367677020752496441044566935490666701848819994695873528056638957197400663802988967689301303582936

Offset: 1

Author

Clark Kimberling, Mar 22 2016

Keywords

• nonn
• frac
• easy

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.

Examples

			sqrt(2) - 1 = 1/3 + 1/(2*7) + 1/(3*36) + 1/(5*1040) + ...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..11
• Eric Weisstein's World of Mathematics, Egyptian Fraction
• Index entries for sequences related to Egyptian fractions

Crossrefs

Cf. A269993, A000045, A188582.

Programs

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 = Sqrt[2] - 1; Table[n[x, k], {k, 1, z}]

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=sqrt(2)-1) = ceil(r(k)/f(k-1, x)); \\ Michel Marcus, Mar 22 2016