%I #32 Feb 06 2017 20:15:28
%S 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,
%T 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,
%U 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
%N Decimal expansion of constant 1.287194... related to a conjectural Viète-like formula for Pi.
%C 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].
%H Sanjar Abrarov, <a href="/A282089/b282089.txt">Table of n, a(n) for n = 1..104</a>
%H S. M. Abrarov and B. M. Quine, <a href="https://arxiv.org/abs/1702.00901">A set of the Viète-like recurrence relations for the unity constant</a>, arXiv:1702.00901 [math.GM], 2017.
%F 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).
%e 1.287194036067924017020927807581...
%t Clear[a, b, c]
%t a[k_] := N[Nest[Sqrt[2 + #1] &, 0, k], 100]
%t b[k_] := b[k] = Sqrt[2 - a[k]]/a[k + 1]
%t c[1] := b[1] = b[1]
%t c[k_] := c[k] = (c[k - 1] + b[k])/(1 - c[k - 1]*b[k])
%t k := 90
%t Print["Index k = ", k]
%t m := 1
%t Print["Power m = ", m]
%t (* The equation (12) *)
%t apprPi := 2^(k + 1)*(1 - c[k]^m)
%t Print["Actual value of Pi is ", N[Pi, 30]]
%t Print["At k = ", k, " the approximated value of Pi is ", N[apprPi, 30]]
%t K := 300
%t Print["Truncating integer K = ", K]
%t Print["Computing the digits ..."]
%t RealDigits[N[Sum[1 - c[k]^m, {k, 1, K}], 30]][[1]]
%Y Cf. A000796.
%K nonn,cons
%O 1,2
%A _Sanjar Abrarov_, Feb 06 2017