A220397 A modified Engel expansion of sqrt(2).

1, 3, 6, 4, 2, 2, 4, 6, 23, 66, 108, 7738, 290, 9, 24, 32, 30, 4, 6, 3, 6, 24, 22, 2, 6, 20, 6, 9, 16, 5, 12, 4, 12, 22, 5, 8, 3, 6, 4, 2, 2, 4, 6, 2, 2, 2, 2, 13, 24, 2, 3, 4, 2, 2, 2, 2, 23, 44, 21, 40, 8, 14, 3, 6, 12, 10, 11, 30, 4, 4, 9, 4, 3, 4, 2, 16, 45, 46, 528

Offset: 1

Author

Peter Bala, Dec 13 2012

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

Cf. A028254, A220335, A220336, A220337, A220338, A220393, A220394, A220395, A220396, A220398.

Formula

Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x). Let x = sqrt(2) - 1. Then a(1) = 1, a(2) = ceiling(1/x) and, for n >= 1, a(n+2) = 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 sqrt(2) = Sum_{n>=1} 1/P(n) = 1 + 1/3 + 1/(3*6) + 1/(3*6*4) + 1/(3*6*4*2) + 1/(3*6*4*2*2) + .... For n >= 2, the error made in truncating this series to n terms is less than the n-th term.