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- #!/usr/bin/ruby
- #
- ## Viète-like formulas:
- #
- # a(c) = {
- # f(0) = 0
- # f(n) = sqrt(c + f(n-1))
- # d = (1 + sqrt(1 + 4*c))/2
- # Limit_{n -> Infinity} (2*d)^n * d - (2*d)^n * f(n)
- # }
- # Closed-forms:
- # a(1) = 2*P where P is the Paris constant (https://oeis.org/A105415)
- # a(2) = pi^2/4
- # See also:
- # https://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula
- local Number!PREC = 1024.numify
- const n = 100
- func a(c) {
- func f(n) {
- n == 0 ? 0 : sqrt(c + f(n-1))
- }
- var d = (1 + sqrt(1 + 4*c))/2
- var t = ((2*d)**n * d - ((2*d)**n * f(n)))
- return t
- }
- for c in (1..10) {
- say ("a(#{c}) = ", a(c).as_dec(48))
- }
- __END__
- a(1) = 2.19728392878831297146933783468719242174669679222
- a(2) = 2.46740110027233965470862274996903778382842485181
- a(3) = 2.72432397395262808042738254231361436370470931728
- a(4) = 2.95697498038250320240181341327815809572441049058
- a(5) = 3.16942068660729603631214225411774841781355055152
- a(6) = 3.36565753974384094582778152437432384197290834457
- a(7) = 3.54871470954831491531526350481479738940975906022
- a(8) = 3.72084501177716618448006573526170234977853589736
- a(9) = 3.88375012406073194374765723289729464351431116217
- a(10) = 4.03874313819098595611125546305106699051058830246
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