A199658 -id:A199658 - cp's OEIS frontend

A191642 Kochański's (or Kochanski's) sequence.

15, 4697, 5548, 14774, 33696, 61072, 111231, 115985, 173819, 563316, 606004, 1751458, 1952544, 3046715, 4397195, 45051907, 653475595, 734915444, 1241384578, 2438767174, 2557084119, 5090226634, 6088149715, 18483120028, 44254634530, 48502484589, 70835215004

Offset: 1

Henryk Fuks, Jun 09 2011

nonn

The sequence of "genitores" used to generate approximants of Pi.

A. A. Kochański, Observationes cyclometricae ad facilitandam praxin accomodatae, Acta Eruditorum 4 (1685) 394-398.

Klaus Brockhaus, Table of n, a(n) for n = 1..1000
Henryk Fuks, Adam Adamandy Kochanski's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 2011-2014; Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.
Henryk Fukś, Magic Squares of Subtraction of Adam Adamandy Kochański, in Research in History and Philosophy of Mathematics, Proceedings of the Canadian Society for History and Philosophy of Mathematics (CSHPM), 2017, pp. 81-95.
Adam Adamany Kochański, Observationes Cyclometricae ad facilitandam Praxin accomodatae, original Latin text from Acta Eruditorum 4, 394-396 (1685), with English translation and annotations (by Henryk Fuks); arXiv:1106.1808 [math.HO], 2011.
Wikipedia, Adam Adamandy Kochański

Cf. A199657, A199658, A199671, A199672.

Digits := 100;
alpha:=Pi;
a:= floor(alpha);
g:=(R,S)->floor( (alpha-a)/(R-alpha*S));
S[1]:=floor(1/(alpha-a));
R[1]:=1+a*S[1];
for n from 2 to 10 do
S[n] := S[n-1]*(g(R[n-1], S[n-1])+1)+1:
R[n] := R[n-1]*(g(R[n-1], S[n-1])+1)+a:
end do:
seq(g(R[i], S[i]), i = 1 .. 10);

g[x_, y_] = Floor[N[(Pi - 3)/(x - Pi*y), 200]];
R = 22; S = 7;
Reap[For[i = 1, i <= 27, i++, b = g[R, S]; S = S*(b+1)+1; R = R*(b+1)+3; Print[b]; Sow[b]]][[2, 1]]; (* Jean-François Alcover, Feb 21 2019, from PARI *)

default(realprecision, 1000);
g(x,y)=floor( (Pi-3)/(x-Pi*y))
R=22; S=7; for(i=1,35, b=g(R,S); S=S*(b+1)+1; R=R*(b+1)+3; print1(b,", "))

I added the unaccented version of the name to the definition, to make it easier to search for. - N. J. A. Sloane, Jan 12 2012

A199657 Numerators of lower rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.

Original entry on oeis.org

25, 333, 1667438, 9252915567, 136727214560643, 4607472064276325091, 281395884679127288508771, 31300458157678523147391901818, 3630416277654441522583270655032758, 631040767628866632706111841438119582182, 355477406146830706663807382201012685829049871, 215421112450033407479085892668138597831784081541979

Offset: 1

Jonathan Vos Post, Nov 08 2011

The corresponding denominators are given in A199658.

The reconstruction refers to the calculation of the "genitores" in A191642, for which Kochański only announced that he would describe them in more detail in a future work: "I will explain the aforementioned method more completely in Polymathic thoughts and inventions, which work, if God prolongs my life, I have decided to put out for public benefit" (translation from Latin by H. Fukś).

			a(1) = 25 because Kochański's first lower bound was 25/8 = a(1)/A199658(1) and his first upper bound was 22/7 = A199671(1)/A199672(1).
a(2) = R(1) * A191642(1) + 3 = 22*15 + 3 = 330 + 3 = 333,
R(2) = R(1) * (A191642(1) + 1 ) + 3 = 22*(15 + 1) + 3 = 355 = A199671(2).

Henryk Fukś, Observationes Cyclometricae by Adam Adamandy Kochański - Latin text with annotated English translation, arXiv:1106.1808 [math.HO] 9 Jun 2011.
Henryk Fukś, Adam Adamandy Kochański's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 2011. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.

Cf. A191642, A199658, A199671, A199672.

a(1) = 25; R(1) = A199671(1) = 22;
a(n) = R(n-1)*A191642(n-1) + 3, where A191642 are Kochański's "genitores";
R(n) = R(n-1)*(A191642(n-1) + 1) + 3;

More terms from Hugo Pfoertner, Mar 07 2020

A199671 Numerators of upper rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.

Original entry on oeis.org

22, 355, 1667793, 9254583360, 136736469144003, 4607608800745469094, 281400492287928033977865, 31300739558170811075425879683, 3630447578393999693394346080912441, 631044398076445026705805235784200494623, 355478037191228783108834088006248470029544494, 215421467928070598707869001502226604080254111086473

Offset: 1

Jonathan Vos Post, Nov 08 2011

The corresponding denominators are given in A199672.

See A199657 for more information and references.

			a(1) = 22 because Kochański's first lower bound was 25/8 = A199657/A199658(1) and his first upper bound was 22/7 = a(1)/A199672(1).
a(2) = a(1) * (A191642(1) + 1) + 3 = 22*(15 + 1) + 3 = 352 + 3 = 355,
a(3) = a(2) * (A191642(2) + 1) + 3 = 355*(4697 + 1) + 3 = 1667793,
a(4) = a(3) * (A191642(3) + 1) + 3 = 1667793*(5548 + 1) + 3 = 9254583360.

Henryk Fukś, Adam Adamandy Kochański's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 2011. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.

Cf. A191642, A199657, A199658, A199672.

g[x_, y_] = Floor[N[(Pi - 3)/(x - Pi*y), 200]];
R = 22; S = 7;
Reap[Print[R]; Sow[R]; For[i = 1, i <= 4, i++, b = g[R, S]; S = S*(b+1)+1; R = R*(b+1)+3; Print[R]; Sow[R]]][[2, 1]] (* Jean-François Alcover, Feb 21 2019 *)

a(1) = 22;

a(n) = a(n-1)*(A191642(n-1) + 1) + 3, where A191642 are Kochański's "genitores".

More terms from Hugo Pfoertner, Mar 07 2020

A199672 Denominators of upper rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.

Original entry on oeis.org

7, 113, 530875, 2945825376, 43524569930401, 1466647432944722498, 89572558672233037120355, 9963334846229825184971327361, 1155607355474812503904084375292947, 200867670528631909428607946113420047541, 113152173559177341323595142380773440920653498, 68570782937729264728805274258460609065120623055491

Offset: 1

Jonathan Vos Post, Nov 08 2011

The corresponding numerators are given in A199671.

See A199657 for more information and references.

			a(1) = 7 because Kochański's first lower bound was 25/8 = A199657(1)/A199658(1) and his first upper bound was 22/7 = A199671(1)/a(1).
a(2) = a(1) * (A191642(1) + 1) + 1 = 7*(15 + 1) + 1 = 112 + 1 = 113,
a(3) = a(2) * (A191642(2) + 1) + 1 = 113*(4697 + 1) + 1 = 530875,
a(4) = a(3) * (A191642(3) + 1) + 1 = 530875*(5548 + 1) + 1 = 2945825376.

Henryk Fukś, Adam Adamandy Kochański's approximations of pi: reconstruction of the algorithm, arXiv preprint arXiv:1111.1739 [math.HO], 2011. Math. Intelligencer, Vol. 34 (No. 4), 2012, pp. 40-45.

Cf. A191642, A199657, A199658, A199671 (numerators).

a(1) = 7;

a(n) = a(n-1)*(A191642(n-1) + 1) + 1, where A191642 are Kochański's "genitores".

More terms from Hugo Pfoertner, Mar 07 2020

A221185 Decimal expansion of sqrt(120-18*sqrt(3))/3.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jan 23 2013

An approximation for Pi, obtained by a geometrical construction by Kochański (1685). - Amiram Eldar, Sep 12 2022

			3.1415333387050946186363982219646240711991241792133632642429...

Benjamin Bold, Famous Problems of Geometry and How to Solve Them, New York: Dover, 1982, p. 44.
J. L. Heilbron, Geometry Civilized: History, Culture, and Technique, Oxford University Press, 2000, pp. 250-252.
Hugo Steinhaus, Mathematical Snapshots, 3rd ed., New York: Dover, 1999, p. 143.

Mordechai Ben-Ari, Mathematical Surprises, Springer, 2022, p. 30.
Henryk Fukś, Adam Adamandy Kochanski's approximations of Pi: reconstruction of the algorithm, Math. Intelligencer, Vol. 34, No. 4 (2012), pp. 40-45; arXiv preprint, arXiv:1111.1739 [math.HO], 2011-2014.
Adam Adamandy Kochański, Observationes Cyclometricae ad facilitandam Praxin accomodatae, Acta Eruditorum, Vol. 4 (1685), pp. 394-398.
Eric Weisstein's World of Mathematics, Kochanski's Approximation.

Cf. A199657, A199658, A199671, A199672.

RealDigits[Sqrt[120 - 18*Sqrt[3]]/3, 10, 100][[1]] (* Amiram Eldar, Sep 12 2022 *)

sqrt(40/3-2*sqrt(3)) \\ Charles R Greathouse IV, Mar 25 2014

cp's OEIS Frontend

A191642 Kochański's (or Kochanski's) sequence.

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A199657 Numerators of lower rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.

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A199671 Numerators of upper rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A199672 Denominators of upper rational approximants of Pi with the first 5 terms given by Adam Adamandy Kochański in 1685, continued using a reconstruction by Fukś that is highly likely to match Kochański's incompletely published method.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A221185 Decimal expansion of sqrt(120-18*sqrt(3))/3.

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI