factorial_expansion.sf

#!/usr/bin/ruby

# Generate the factorial expansion of a given real constant.

# Such that:
#   x = Sum_{k>=0} f(k) / k!

# See also:
#   https://oeis.org/A067882
#   https://oeis.org/A096622

func f(n, x, r=:floor) {
    return x.(r) if (n == 0)
    (x * n!).(r) - n*((n-1)! * x).(r)
}

var x = Num.EulerGamma

var a = 30.of { f(_, x) }
var b = 30.of { f(_, x, :round) }

say "const: #{x}"

say ("floor: ", a.len.range.sum { |k|
    a[k] / k!
})

say ("round: ", b.len.range.sum { |k|
    b[k] / k!
})

say ''
say "floor: #{a}"
say "round: #{b}"

__END__
const: 0.57721566490153286060651209008240243104215933594
floor: 0.577215664901532860606512090082341769080234288111
round: 0.57721566490153286060651209008245486870909876528

floor: [0, 0, 1, 0, 1, 4, 1, 4, 1, 3, 0, 2, 3, 0, 5, 14, 12, 16, 14, 7, 13, 18, 17, 19, 11, 22, 13, 13, 26, 12]
round: [1, 0, -1, 0, 2, -1, 2, -3, 1, 3, 0, 2, 3, 0, 6, 0, -3, 0, -4, 8, -6, -2, -4, -4, 12, -2, -12, -13, -2, 13]