A063673 -id:A063673 - cp's OEIS frontend

A002486 Apart from two leading terms (which are present by convention), denominators of convergents to Pi (A002485 and A046947 give numerators).

1, 0, 1, 7, 106, 113, 33102, 33215, 66317, 99532, 265381, 364913, 1360120, 1725033, 25510582, 52746197, 78256779, 131002976, 340262731, 811528438, 1963319607, 4738167652, 6701487259, 567663097408, 1142027682075, 1709690779483, 2851718461558, 44485467702853

Offset: 0

N. J. A. Sloane

Disregarding first two terms, integer diameters of circles beginning with 1 and a(n+1) is the smallest integer diameter with corresponding circumference nearer an integer than is the circumference of the circle with diameter a(n). See PARI program. - Rick L. Shepherd, Oct 06 2007

a(n+1) = numerator of fraction obtained from truncated continued fraction expansion of 1/Pi to n terms. - Artur Jasinski, Mar 25 2008

			The convergents are 3, 22/7, 333/106, 355/113, 103993/33102, ...

P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 171 (but beware errors).
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 293.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 274.

Daniel Mondot, Table of n, a(n) for n = 0..1947 (terms 0..201 from T. D. Noe, terms 202..1000 from G. C. Greubel).
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
Marc Daumas, Des implantations différentes ..., see p. 8. [Broken link]
P. Finsler, Über die Faktorenzerlegung natuerlicher Zahlen, Elemente der Mathematik, 2 (1947), 1-11, see p. 7.
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.
G. P. Michon, Continued Fractions
Eric Weisstein's World of Mathematics, Pi.
Eric Weisstein's World of Mathematics, Pi Continued Fraction
Eric Weisstein's World of Mathematics, Pi Approximations
Index entries for sequences related to the number Pi

Cf. A002485 (numerators), A072398/A072399, A063674/A063673, A132049/A132050.

Digits := 60: E := Pi; convert(evalf(E),confrac,50,'cvgts'): cvgts;
with(numtheory):cf := cfrac (Pi,100): seq(nthdenom (cf,i), i=-2..28 ); # Zerinvary Lajos, Feb 07 2007

Join[{1,0},Denominator[Convergents[Pi,30]]] (* Harvey P. Dale, Sep 13 2013 *)

for(i=1,#cf=contfrac(Pi),print1(contfracpnqn(vecextract(cf,2^i-1))[2,2]",")) \\ M. F. Hasler, Apr 01 2013

Extended and corrected by David Sloan, Sep 23 2002

A063674 Numerators of increasingly better rational approximations to Pi with increasing denominators (3/1, 13/4, 16/5, 19/6, 22/7, 179/57, ...)

Original entry on oeis.org

3, 13, 16, 19, 22, 179, 201, 223, 245, 267, 289, 311, 333, 355, 52163, 52518, 52873, 53228, 53583, 53938, 54293, 54648, 55003, 55358, 55713, 56068, 56423, 56778, 57133, 57488, 57843, 58198, 58553, 58908, 59263, 59618, 59973, 60328, 60683, 61038, 61393, 61748

Offset: 1

Suren L. Fernando (fernando(AT)truman.edu), Jul 27 2001

Numerators of the sequence (3/1, 13/4, 16/5, 19/6, 22/7, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106, 355/113, 52163/16604, 52518/16717, ...)

Large jumps occur after the classical approximations 22/7 and 355/113, which are sufficiently precise to require a much larger denominator for a better approximation. - M. F. Hasler, Apr 01 2013

P. D. Howard, Table of n, a(n) for n = 1..18865
Jean-Louis Sikorav, Best rational approximations of an irrational number, arXiv:1807.06284 [math.NT], 2018.

Cf. A063673 (denominators), A057082, A002485/A002486, A132049/A132050, A072398/A072399.

piapprox[n_] := Block[{a, i}, a = {3/1}; For[i = 2, i <= n, i++, If[Abs[Round[i Pi]/i - Pi] < Abs[Last[a] - Pi], AppendTo[a, Round[i Pi]/i], Null]]; Return[a]] (* Suren Fernando via Alexander R. Povolotsky, Aug 03 2008 *)

{e=1; for(d=1,1e5, abs( Pi-round(Pi*d)/d ) < e & !print1(round(Pi*d)",") & e=abs(Pi - round(Pi*d)/d))} \\ [M. F. Hasler, Apr 01 2013]

More terms from M. F. Hasler, Apr 01 2013

A132050 Denominator of 2nA000111(n-1)/A000111(n): approximations of Pi using Euler (up/down) numbers.

Original entry on oeis.org

1, 1, 1, 5, 8, 61, 136, 1385, 3968, 50521, 176896, 2702765, 260096, 199360981, 951878656, 19391512145, 104932671488, 2404879675441, 14544442556416, 74074237647505, 2475749026562048, 69348874393137901, 507711943253426176

Offset: 1

Wolfdieter Lang, Sep 14 2007

The rationals r(n)=2*n*e(n-1)/e(n), where e(n)=A000111(n), approximate Pi as n -> oo. - M. F. Hasler, Apr 03 2013
Numerators are given in A132049.
See the Delahaye reference and a link by W. Lang given in A132049.
From Paul Curtz, Mar 17 2013: (Start)
Apply the Akiyama-Tanigawa transform (or algorithm) to A046978(n+2)/A016116(n+1):
1,        1/2,      0,   -1/4,  -1/4,  -1/8,      0, 1/16, 1/16;
1/2,        1,    3/4,      0,  -5/8,  -3/4,  -7/16,    0;  = Balmer0(n)
-1/2,     1/2,    9/4,    5/2,   5/8, -15/8, -49/16;
-1,      -7/2,   -3/4,   15/2,  25/2,  57/8;
5/2,    -11/2,  -99/4,    -20, 215/8;
8,       77/2,  -57/4, -375/2;
-61/2,  211/2, 2079/4;
-136, -1657/2;
1385/2;
The first column is PIEULER(n) = 1, 1/2, -1/2, -1, 5/2, 8, -61/2, -136, 1385/2,... = c(n)/d(n). Abs c(n+1)=1,1,1,5,8,61,... =a(n) with offset=1.
For numerators of Balmer0(n) see A076109, A000265 and A061037(n-1) (End).
Other completely unrelated rational approximations of Pi are given by A063674/A063673 and other references there. - M. F. Hasler, Apr 03 2013

			Rationals r(n): [2, 4, 3, 16/5, 25/8, 192/61, 427/136, 4352/1385, 12465/3968, 158720/50521, ...].

Cf. triangle A008281 (main diagonal give zig-zag numbers A000111).

e[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1)*(2^(n + 1) - 1)*BernoulliB[n + 1])/(n + 1)]]; r[n_] := 2*n*(e[n - 1]/e[n]); a[n_] := Denominator[r[n]]; Table[a[n], {n, 1, 23}] (* Jean-François Alcover, Mar 26 2013 *)

from itertools import count, islice, accumulate
from fractions import Fraction
def A132050_gen(): # generator of terms
    yield 1
    blist = (0,1)
    for n in count(2):
        yield Fraction(2*n*blist[-1],(blist:=tuple(accumulate(reversed(blist),initial=0)))[-1]).denominator
A132050_list = list(islice(A132050_gen(),40)) # Chai Wah Wu, Jun 09-11 2022

a(n)=denominator(r(n)) with the rationals r(n):=2*n*e(n-1)/e(n) where e(n):=A000111(n).

Definition made more explicit, and initial terms a(1)=a(2)=1 added by M. F. Hasler, Apr 03 2013

A327360 Minimal numerator among the fractions with n-digit numerator and n-digit denominator that best approximate Pi.

Original entry on oeis.org

3, 44, 355, 3195, 99733, 833719, 5419351, 80143857, 657408909, 6167950454, 42106686282, 983339177173, 8958937768937, 94960529682104, 428224593349304, 6134899525417045, 66627445592888887, 430010946591069243, 5293386250278608690, 31760317501671652140

Offset: 1

Jason Zimba, Sep 03 2019

			The fractions with 2-digit numerators and 2-digit denominators that best approximate Pi are 44/14 and 88/28. The fraction with 6-digit numerator and 6-digit denominator that best approximates Pi is 833719/265381.

O. Zelenyak, Programming workshop on Turbo Pascal: Tasks, Algorithms and Solutions, Litres, 2018, page 255. (Provides first 8 terms. Also contains similar sequences for sqrt(2) and e.)

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000
Jon E. Schoenfield, Magma program
O. Zelenyak, Programming workshop on Turbo Pascal: Tasks, Algorithms and Solutions, Litres, 2018, page 255.

A327361 gives the corresponding denominators.
Cf. A072398/A072399, which gives the best rational approximation to Pi subject to a different constraint.
Cf. A002485/A002486, A063674/A063673, A325158/A325159.

(* Given the 8th term, find the 9th term *)
(* This took twelve-plus hours to run on a laptop *)
ResultList = {};
nVal = 9;
tol = Abs[80143857/25510582 - Pi]; (* 80143857 is A327360(8), 25510582 is A327361(8) *)
Do[
  CurrentNumerator = i;
  Do[
   CurrentDenominator = j;
   CurrentQuotient = N[CurrentNumerator/CurrentDenominator];
   If[
    Abs[CurrentQuotient - Pi] <= tol,
    ResultList = Append[ResultList, {CurrentNumerator, CurrentDenominator}]
    ],
   {j, Floor[i/(Pi + tol)], Floor[i/(Pi - tol)] + 1}],
  {i, Floor[(Pi - tol)*10^(nVal - 1)], (10^nVal - 1)}];
DifferenceList =
  Table[
   Abs[ResultList[[i, 1]]/ResultList[[i, 2]] - Pi],
   {i, 1, Length[ResultList]}];
Extract[ResultList, Position[DifferenceList, Min[DifferenceList]]]

Terms a(10) and beyond from Jon E. Schoenfield, Mar 11 2021

A327361 Minimal denominator among the fractions with n-digit numerator and n-digit denominator that best approximate Pi.

Original entry on oeis.org

1, 14, 113, 1017, 31746, 265381, 1725033, 25510582, 209259755, 1963319607, 13402974518, 313006581566, 2851718461558, 30226875395063, 136308121570117, 1952799169684491, 21208174623389167, 136876735467187340, 1684937174853026414, 10109623049118158484

Offset: 1

Jason Zimba, Sep 03 2019

			The fractions with 2-digit numerators and 2-digit denominators that best approximate Pi are 44/14 and 88/28.
The fraction with 6-digit numerator and 6-digit denominator that best approximates Pi is 833719/265381.

O. Zelenyak, Programming workshop on Turbo Pascal: Tasks, Algorithms and Solutions, Litres, 2018, page 255. (Provides first 8 terms. Also contains similar sequences for sqrt(2) and e.)

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000
O. Zelenyak, Programming workshop on Turbo Pascal: Tasks, Algorithms and Solutions, Litres, 2018, page 255.

A327360 gives the corresponding numerators.
Cf. A072398/A072399, which gives the best rational approximation to Pi subject to a different constraint.
Cf. A002485/A002486, A063674/A063673, A325158/A325159.

(* Given the 8th term, find the 9th term *)
(* This took twelve-plus hours to run on a laptop *)
ResultList = {};
nVal = 9;
tol = Abs[80143857/25510582 - Pi]; (* 80143857 is A327360(8), 25510582 is A327361(8) *)
Do[
  CurrentNumerator = i;
  Do[
   CurrentDenominator = j;
   CurrentQuotient = N[CurrentNumerator/CurrentDenominator];
   If[
    Abs[CurrentQuotient - Pi] <= tol,
    ResultList = Append[ResultList, {CurrentNumerator, CurrentDenominator}]
    ],
   {j, Floor[i/(Pi + tol)], Floor[i/(Pi - tol)] + 1}],
  {i, Floor[(Pi - tol)*10^(nVal - 1)], (10^nVal - 1)}];
DifferenceList =
  Table[
   Abs[ResultList[[i, 1]]/ResultList[[i, 2]] - Pi],
   {i, 1, Length[ResultList]}];
Extract[ResultList, Position[DifferenceList, Min[DifferenceList]]]

a(10)-a(20) from Jon E. Schoenfield, Mar 12 2021

A224365 a(n) = A063674(n+1) - A063674(n).

Original entry on oeis.org

10, 3, 3, 3, 157, 22, 22, 22, 22, 22, 22, 22, 22, 51808, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355, 355

Offset: 1

Paul Curtz, Apr 09 2013

The repeated terms (3, 22, 355, 5419351, ... from A063674) are the numerators of fractions (3/1, 22/7, 355/113, 5419351/1725033, ...) leading to Pi.
Zu Chongzhi (5th century) discovered 22/7 and 355/113. Adriaan Anthonisz Metius rediscovered 355/113 in 1585.
First differences of A063673 give the denominators: 3, 1, 1, 1, 50, 7, 7, 7, 7, 7, 7, 7, 7, 16489, 113, 113, ... .
Hence 10/3, 157/50, 51808/16489, ... .

Vincenzo Librandi, Table of n, a(n) for n = 1..168

Cf. A063673, A063674.

Cf. A046947, A072398, A002485. A132049, A003077, A068028, A068079.

A224365 = Reap[ For[ delta0 = 1; d = 1, d < 10^5, d++, delta = Abs[Pi - Round[Pi*d]/d]; If[ delta < delta0, Sow[ Round[Pi*d]]; delta0 = delta]]][[2, 1]] // Differences (* Jean-François Alcover, Apr 10 2013 *)

a(n) = A063674(n+1) - A063674(n).

A346534 Denominators of approximations j/k for Pi such that abs(j/k - Pi)sqrt(5)k^2 < 1.

Original entry on oeis.org

1, 7, 14, 113, 226, 339, 452, 565, 678, 791, 904, 1017, 1130, 1243, 33215, 99532, 364913, 1725033, 3450066, 25510582, 131002976, 340262731, 811528438, 1963319607, 6701487259, 13402974518, 20104461777, 26805949036, 33507436295, 40208923554, 567663097408

Offset: 1

June Richardson, Jul 22 2021

Define two parameters E and M for a rational approximation j/k for an irrational number x: E = abs(j/k - x) (the absolute error) and M = 1/(sqrt(5)*k^2). Hurwitz's theorem states that every real number has infinitely many rational approximations that satisfy E/M < 1, making each such approximation a "strong approximation". This sequence lists the denominators of such numbers for the irrational number Pi.

			22/7 ~ 3.1428571 and E/M ~ 0.1385.
355/113 ~ 3.1415929 and E/M ~ 0.0076.
From _Jon E. Schoenfield_, Aug 06 2021: (Start)
    k       j    E = |j/k - Pi|  M = 1/(sqrt(5)*k^2)    E/M
  -----  ------  --------------  -------------------  -------
      1       3  0.141592653590  0.44721359549995794  0.31661
      7      22  0.001264489267  0.00912680807142771  0.13855
     14      44  0.001264489267  0.00228170201785693  0.55419
    113     355  0.000000266764  0.00003502338440755  0.00762
    226     710  0.000000266764  0.00000875584610189  0.03047
    339    1065  0.000000266764  0.00000389148715639  0.06855
    452    1420  0.000000266764  0.00000218896152547  0.12187
    565    1775  0.000000266764  0.00000140093537630  0.19042
    678    2130  0.000000266764  0.00000097287178910  0.27420
    791    2485  0.000000266764  0.00000071476294709  0.37322
    904    2840  0.000000266764  0.00000054724038137  0.48747
   1017    3195  0.000000266764  0.00000043238746182  0.61696
   1130    3550  0.000000266764  0.00000035023384408  0.76167
   1243    3905  0.000000266764  0.00000028944945791  0.92163
  33215  104348  0.000000000332  0.00000000040536522  0.81810
(End)

AMS, Rational approximation of irrational numbers
Jon E. Schoenfield, Magma program with explanation of algorithm
Wikipedia, Hurwitz's theorem (number theory)

Cf. A000796, A001203, A063673.

Cf. A002163 (sqrt(5)).

Magma
```
// See Links.
```

a={}; For[k=1,k<=10^6,k++,If[Abs[Round[k Pi]/k-Pi]Sqrt[5] k^2<1,AppendTo[a,k]]]; a (* Stefano Spezia, Aug 07 2021 *)

is(k) = my(j=round(Pi*k)); abs(j/k - Pi)*sqrt(5)*k^2 < 1; \\ Jinyuan Wang, Aug 06 2021

a(17)-a(19) from Jinyuan Wang, Aug 06 2021

a(20)-a(31) from Jon E. Schoenfield, Aug 06 2021

cp's OEIS Frontend

A002486 Apart from two leading terms (which are present by convention), denominators of convergents to Pi (A002485 and A046947 give numerators).

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A063674 Numerators of increasingly better rational approximations to Pi with increasing denominators (3/1, 13/4, 16/5, 19/6, 22/7, 179/57, ...)

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A132050 Denominator of 2*n*A000111(n-1)/A000111(n): approximations of Pi using Euler (up/down) numbers.

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A327360 Minimal numerator among the fractions with n-digit numerator and n-digit denominator that best approximate Pi.

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Mathematica

Extensions

A327361 Minimal denominator among the fractions with n-digit numerator and n-digit denominator that best approximate Pi.

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Mathematica

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A224365 a(n) = A063674(n+1) - A063674(n).

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A346534 Denominators of approximations j/k for Pi such that abs(j/k - Pi)*sqrt(5)*k^2 < 1.

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Magma

A132050 Denominator of 2nA000111(n-1)/A000111(n): approximations of Pi using Euler (up/down) numbers.

A346534 Denominators of approximations j/k for Pi such that abs(j/k - Pi)sqrt(5)k^2 < 1.