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A220396
A modified Engel expansion of the Euler-Mascheroni constant gamma.
5
2, 7, 18, 4, 2, 2, 3, 1466, 1464, 9, 24, 4, 2, 9, 104, 60, 8, 2, 3, 6, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 32, 30, 2, 13, 36, 6, 4, 3, 6, 6, 4, 4, 6, 2, 4, 6, 2, 4, 6, 9, 24, 4, 5, 8, 2, 2, 2, 2, 2, 3, 20
OFFSET
1,1
COMMENTS
See A220393 for the definition of the modified Engel expansion of a positive real number. For further details see the Bala link.
FORMULA
Let h(x) = x*(floor(1/x) + (floor(1/x))^2) - floor(1/x). Let x = gamma (see A001620). Then a(1) = 1 + floor(1/x) and, for n >= 1, a(n+1) = 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) = 1/2 + 1/(2*7) + 1/(2*7*18) + 1/(2*7*18*4) + 1/(2*7*18*4*2) + .... The error made in truncating this series to n terms is less than the n-th term.
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Dec 13 2012
STATUS
approved