generalized_expansion.sf

#!/usr/bin/ruby

# Generalized expansion of a given real constant, given a function t(k).

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

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

local Num!PREC = 300.numify

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

var t = { .superfactorial }

var x = Num.e
var a = 15.of { f(_, x, t) }
var b = 15.of { f(_, x, t, {.round}) }

say "const: #{x}"

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

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

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

__END__
const: 2.71828182845904523536028747135266249775724709369995957496696762772407663035
floor: 2.71828182845904523536028747135266249775724709369995957496696762771492315184
round: 2.71828182845904523536028747135266249775724709369995957496696762772939204445

floor: [2, 0, 1, 2, 14, 103, 590, 1985, 12783, 261726, 3567625, 4152491, 421616733, 2483567501, 21279242709]
round: [3, 0, -1, 3, -9, -16, -130, 1985, 12784, -101153, -61175, 4152492, -57384867, 2483567501, 21279242710]