A282089 Decimal expansion of constant 1.287194... related to a conjectural Viète-like formula for Pi.

1, 2, 8, 7, 1, 9, 4, 0, 3, 6, 0, 6, 7, 9, 2, 4, 0, 1, 7, 0, 2, 0, 9, 2, 7, 8, 0, 7, 5, 8, 1, 1, 9, 8, 7, 6, 4, 4, 0, 8, 3, 5, 4, 3, 5, 6, 6, 9, 9, 2, 7, 8, 0, 5, 4, 4, 8, 6, 1, 4, 1, 2, 9, 3, 2, 7, 1, 4, 5, 2, 8, 3, 9, 1, 4, 4, 8, 7, 2, 0, 2, 2, 1, 1, 2, 3, 7, 9, 0, 7, 9, 9, 2, 6, 0, 9, 3, 4, 0, 3, 3, 9, 9, 8

Offset: 1

Author

Sanjar Abrarov, Feb 06 2017

Keywords

• nonn
• cons

Comments

Conjecture: Pi = lim_{k -> infinity} 2^{k + 1}*(1 - c_k), where the variable c_k is defined by a set of the Viète-like recurrence relations {a_1 = sqrt(2), a_k = sqrt(2 + a_{k - 1}), b_k = sqrt(2 - a_k)/a_{k + 1}, c_1 = b_1, c_k = (c_{k - 1} + b_k)/(1 - c_{k - 1}*b_k)}. From this conjecture it follows that Sum_{k >= 1} (1 - c_k) is convergent [Abrarov and Quine].

Examples

			1.287194036067924017020927807581...
		

Links

• Sanjar Abrarov, Table of n, a(n) for n = 1..104
• S. M. Abrarov and B. M. Quine, A set of the Viète-like recurrence relations for the unity constant, arXiv:1702.00901 [math.GM], 2017.

Crossrefs

Cf. A000796.

Programs

Mathematica

Clear[a, b, c]
a[k_] := N[Nest[Sqrt[2 + #1] &, 0, k], 100]
b[k_] := b[k] = Sqrt[2 - a[k]]/a[k + 1]
c[1] := b[1] = b[1]
c[k_] := c[k] = (c[k - 1] + b[k])/(1 - c[k - 1]*b[k])
k := 90
Print["Index k = ", k]
m := 1
Print["Power m = ", m]
(* The equation (12) *)
apprPi := 2^(k + 1)*(1 - c[k]^m)
Print["Actual value of Pi is ", N[Pi, 30]]
Print["At k = ", k, " the approximated value of Pi is ", N[apprPi, 30]]
K := 300
Print["Truncating integer K = ", K]
Print["Computing the digits ..."]
RealDigits[N[Sum[1 - c[k]^m, {k, 1, K}], 30]][[1]]

Formula

Sum_{k >= 1} (1 - c_k) = 1.287194... , where c_k is computed by the recurrence equations a_1 = sqrt(2), a_k = sqrt(2 + a_{k - 1}), b_k = sqrt(2 - a_k)/a_{k + 1}, c_1 = b_1 and c_k = (c_{k - 1} + b_k)/(1 - c_{k - 1}*b_k).