login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A282089 Decimal expansion of constant 1.287194... related to a conjectural Viète-like formula for Pi. 1
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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].

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.

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).

EXAMPLE

1.287194036067924017020927807581...

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]]

CROSSREFS

Cf. A000796.

Sequence in context: A011060 A278808 A323459 * A195367 A135725 A265295

Adjacent sequences:  A282086 A282087 A282088 * A282090 A282091 A282092

KEYWORD

nonn,cons

AUTHOR

Sanjar Abrarov, Feb 06 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 25 16:01 EST 2022. Contains 350572 sequences. (Running on oeis4.)