A220394 A modified Engel expansion of exp(1).

1, 1, 2, 3, 4, 5, 8, 2, 10, 99, 20, 2, 2, 2, 2, 2, 2, 3, 6, 4, 8, 14, 2, 2, 4, 6, 10, 252, 81, 30, 28, 31, 60, 4, 6, 3, 4, 2, 2, 2, 2, 19, 54, 8, 6, 22, 63, 4, 2, 4, 6, 2, 2, 5, 12, 4, 2, 2, 2, 2, 6, 15, 10, 348, 172, 2, 2, 4, 6, 4, 30, 207, 220

Offset: 1

Author

Peter Bala, Dec 13 2012

Keywords

• nonn
• easy

Comments

See A220393 for a description of the modified Engel expansion of a positive real number. For further details see the Bala link.

The Engel expansion for exp(1) is the sequence of positive integers A000027.

Links

• Peter Bala, A modified Engel expansion
• Wikipedia, Engel Expansion

Crossrefs

Cf. A000027, A220335, A220336, A220337, A220338, A220393, A220395, A220396, A220397, A220398.

Formula

Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x) where x = exp(1) - 2, then a(1) = a(2) = 1, a(3) = ceiling(1/x) and, for n >= 1, a(n+3) = floor(1/h^(n-1)(x))*(1 + floor(1/h^(n)(x))).

Put P(n) = Product_{k = 1..n} a(k). Then we have the Egyptian fraction series expansion exp(1) = Sum_{n>=1} 1/P(n) = 1/1 + 1/1 + 1/2 + 1/(2*3) + 1/(2*3*4) + 1/(2*3*4*5) + 1/(2*3*4*5*8) + .... For n >= 3, the error made in truncating this series to n terms is less than the n-th term.