A046965 -id:A046965 - cp's OEIS frontend

A068028 Decimal expansion of 22/7.

3, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4

Offset: 1

Nenad Radakovic, Mar 22 2002

This is an approximation to Pi. It is accurate to 0.04025%.
Consider the recurring part of 22/7 and the sequences R(i) = 2, 1, 4, 2, 3, 0, 2, ... and Q(i) = 1, 4, 2, 8, 5, 7, 1, .... For i > 0, let X(i) = 10*R(i) + Q(i). Then Q(i+1) = floor(X(i)/Y); R(i+1) = X(i) - Y*Q(i+1); here Y=5; X(0)=X=7. Note 1/7 = 7/49 = X/(10*Y-1). Similar comment holds elsewhere. If we consider the sequences R(i) = 3, 2, 3, 5, 5, 1, 4, 0, 6, 4, 6, 3, 4, 3, 1, 1, 5, 2, 6, 0, 2, 0, 3, ... and Q(i) = A021027, we have X=3; Y=7 (attributed to Vedic literature). - K.V.Iyer, Jun 16 2010, Jun 18 2010
The sequence of convergents of the continued fraction of Pi begins [3, 22/7, 333/106, 355/113, 103993/33102, ...]. 22/7 is the second convergent. The summation 240*Sum_{n >= 1} 1/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)(4*n+6)*(4*n+7)) = 22/7 - Pi shows that 22/7 is an over-approximation to Pi. - Peter Bala, Oct 12 2021

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 187, 239.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi and §13.3 Solving Triangles, pp. 90, 479.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 49.

D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163. - _N. J. A. Sloane_, Mar 24 2012
D. P. Dalzell, On 22/7, J. London Math. Soc. 19, 133-134, 1944.
Dale Winham, Facts about Pi.
Index entries for sequences related to the number Pi.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A068079, A068089, A002485, A002486, A046965, A046947.

I:=[3,1,4,2,8]; [n le 5 select I[n] else Self(n-1)-Self(n-3)+Self(n-4): n in [1..100]]; // Vincenzo Librandi, Mar 27 2015

CoefficientList[Series[(3 - 2 x + 3 x^2 + x^3 + 4 x^4) / ((1 - x) (1 + x) (1 - x + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 27 2015 *)
Join[{3},LinearRecurrence[{1, 0, -1, 1},{1, 4, 2, 8},104]] (* Ray Chandler, Aug 26 2015 *)
RealDigits[22/7,10,120][[1]] (* Harvey P. Dale, Oct 04 2021 *)

a(0)=3, a(n) = floor(714285/10^(5-(n mod 6))) mod 10. - Sascha Kurz, Mar 23 2002 [corrected by Jason Yuen, Aug 18 2024]
Equals 100*A021018 - 4 = 3 + A020806. - R. J. Mathar, Sep 30 2008
For n>1 a(n) = A020806(n-2) (note offset=0 in A020806 and offset=1 in A068028). - Zak Seidov, Mar 26 2015
G.f.: x*(3-2*x+3*x^2+x^3+4*x^4)/((1-x)*(1+x)*(1-x+x^2)). - Vincenzo Librandi, Mar 27 2015

More terms from Sascha Kurz, Mar 23 2002

Alternative to broken link added by R. J. Mathar, Jun 18 2010

A068079 Decimal expansion of 355 / 113.

Original entry on oeis.org

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

Offset: 1

Nenad Radakovic, Mar 22 2002

This is an approximation to Pi. It is accurate to 0.00000849%.
355/113 is the third convergent of the continued fraction expansion of Pi (A001203). - Lekraj Beedassy, Jun 18 2003
In one of Ramanujan's papers, a note at the bottom states that "If the area of the circle be 140,000 square miles, then RD [RD = d/2 * Sqrt(355/113) = r*Sqrt(Pi), very nearly] is greater than the true length by about an inch."
This approximation of Pi was suggested by the astronomer Tsu Chúng-chih (A.D. 430 - 501) (see Gullberg). - Stefano Spezia, Jan 13 2025

			3.141592920353982300884955752212389380530973451327433628318584...

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 88.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 187, 238-239.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi, p. 91.
Ramanujan's papers, "Squaring the circle", Journal of the Indian Mathematical Society, V, 1913, 132. - Robert G. Wilson v, May 30 2014
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 49.

Dario Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163. [_N. J. A. Sloane_, Mar 24 2012]
Dale, Fun and interesting facts about Pi.
Index entries for sequences related to the number Pi.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1).

Cf. A068028, A068089, A002485, A002486, A046965, A046947, A083871.

Digits:=100: evalf(355/113); # Wesley Ivan Hurt, Mar 14 2015

Flatten[RealDigits[355/113, 10, 100]] (* Wesley Ivan Hurt, Mar 14 2015 *)

355/113. \\ Charles R Greathouse IV, May 30 2014

a(n) = if(n==1, 3, digits(16*10^112 \ 113)[(n-2) % 112 + 1]) \\ Jeppe Stig Nielsen, Dec 14 2019

a(n) = a(n - 112) for n > 113. - Jeppe Stig Nielsen, Dec 14 2019

More terms from Sascha Kurz, Mar 23 2002

Terms a(106) and beyond from Jeppe Stig Nielsen, Dec 14 2019

A068089 Decimal expansion of 104348 / 33215.

Original entry on oeis.org

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

Offset: 1

Nenad Radakovic, Mar 22 2002

This is an approximation to Pi. It is accurate to 0.00000001056%.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Dale, Facts about Pi
Index entries for sequences related to the number Pi
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A002485, A002486, A046965, A046947, A068028, A068079.

First[RealDigits[104348/33215,10,100]] (* Paolo Xausa, Nov 07 2023 *)

More terms from Sascha Kurz, Mar 23 2002

A046959 Numbers k where sin(k) increases monotonically to 1 (or cosec(k) decreases).

Original entry on oeis.org

0, 1, 2, 8, 14, 33, 322, 366, 699, 1409, 2119, 2829, 3539, 4249, 4959, 5669, 6379, 7089, 7799, 8509, 9219, 9929, 10639, 11349, 12059, 12769, 13479, 14189, 14899, 15609, 16319, 17029, 17739, 18449, 19159, 19869, 20579, 21289, 21999, 22709, 23419, 24129, 24839

Offset: 1

Olivier Gérard

nonn

sin(37362253) = 0.9999999999999996492283960073698922883715161030254...

Hugo Pfoertner, Table of n, a(n) for n = 1..99 (first 88 terms from Robert G. Wilson v)

Cf. A046964, A046965.

d = -1; lst = {}; Do[a = Sin@n; If[a > d, d = a; Print@n; AppendTo[lst, n]], {n, 0, 111111111}]; lst (* Robert G. Wilson v, Aug 24 2007 *)
DeleteDuplicates[Table[{n,Sin[n]},{n,0,25000}],GreaterEqual[#1[[2]],#2[[2]]]&] [[All,1]] (* Harvey P. Dale, May 14 2022 *)

A046964 Sin(n) decreases monotonically to -1.

Original entry on oeis.org

1, 3, 4, 5, 11, 344, 1054, 1764, 2474, 3184, 3894, 4604, 5314, 6024, 6734, 7444, 8154, 8864, 9574, 10284, 10994, 11704, 12414, 13124, 13834, 14544, 15254, 15964, 16674, 17384, 18094, 18804, 19514, 20224, 20934, 21644, 22354, 23064, 23774, 24484, 25194, 25904

Offset: 1

Wouter Meeussen

nonn

Sin(10265498) =-0.9999999999999999313932793053103935998520142594607...

Hugo Pfoertner, Table of n, a(n) for n = 1..93, terms a(1)..a(88) from Robert G. Wilson v

Cf. A046959, A046965.

z={}; current=1; Timing[ Do[ If[ Sin[ n ] d, d = a; Print@n; AppendTo[lst, n]], {n, 111111111}]; lst (* Robert G. Wilson v, Aug 24 2007 *)

d=oo; print1("1, 3, "); for(k=1,10^8, my(di=2*k/Pi,dir=round(di),dd);if(dir%4==3,dd=abs(di-dir); if(ddHugo Pfoertner, Feb 29 2020

A337371 Integers k with abs(sin(k)) < 1/k.

Original entry on oeis.org

1, 3, 22, 44, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 312689, 1146408, 5419351, 10838702, 6167950454, 21053343141, 42106686282, 63160029423, 84213372564, 105266715705, 8958937768937, 17917875537874, 428224593349304, 856449186698608, 6134899525417045

Offset: 1

Anian Brosig, Aug 25 2020

nonn

The values > 1 appear to be a subset of the numerators of continued fractions of Pi (A002485) (and/or Pi/2: A096456) and their multiples. Is it possible to find a term k here but not in |A332095| (k |tan k| < 1)? - M. F. Hasler, Oct 09 2020

Cf. A092328, A332095, A046965, A088306, A337249, A332095.

Select[Range[3200], Abs[Sin[#]] < 1/# &] (* Amiram Eldar, Aug 25 2020 *)

print1(1);apply( n-> forstep(n=n,oo,n,abs(sin(n))<1/n||return; print1(","n)), contfracpnqn(c=contfrac(Pi),#c)[1,]); \\ M. F. Hasler, Oct 09 2020

import numpy as np
for x in range(1, 10**9):
    if np.abs(np.sin(x)) < 1/x:
        print(x, end=", ")

More terms from M. F. Hasler, Oct 09 2020

A172451 a(1) = 1, and for each n >= 2, a(n) is the smallest number such that 1/sin(a(n)) < 1/sin(a(k)) for all k < n, so that 1/sin(a(1)) > 1/sin(a(2)) > ... > 1/sin(a(n)) > ...

Original entry on oeis.org

1, 2, 4, 6, 22, 333, 355, 103993, 104348, 1042060, 1146408, 4272943, 5419351, 80143857

Offset: 1

Michel Lagneau, Feb 03 2010

			1/sin(1) = 1.1883951; 1/sin(2) = 1.0997501; 1/sin(4) = - 1.3213487.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 29, Ellipses, Paris 2008. Also Entry 137, p. 47.

Eric Weisstein's World of Mathematics, Pi

Cf. A002485, A046959, A046965, A046964, A172445, A172446.

a:= evalf(1/sin(1)); for n from 2 to 10000000 do; if a > evalf(1/sin(n)) then a:= evalf(1/sin(n)); print(n); else fi ; od;

vm = 2; s = {}; Do[v = 1/Sin[n]; If[v < vm, vm = v; AppendTo[s, n]], {n, 1, 110000}]; s (* Amiram Eldar, Aug 10 2019 *)

lista(NN) = {my(x=2); for(k=1, NN, if(1/sin(k)Jinyuan Wang, Aug 12 2019

a(13) corrected and a(14) added by Amiram Eldar, Aug 10 2019

A385636 Numbers k where sin(sqrt(k)) reaches successive record high values.

Original entry on oeis.org

0, 1, 2, 60, 61, 62, 200, 417, 2687, 1536009, 57157270, 81436635, 143005063, 706132910, 2016118929

Offset: 1

Artur Jasinski, Jul 05 2025

Each record value is ever closer to 1, but never reaches 1 since Pi is not algebraic.

			         n sin(sqrt(n))
         0 0.00000000000000000000000000000000000000
         1 0.84147098480789650665250232163029899962
         2 0.98776594599273552706913407207894265590
        60 0.99417205583623141588832072787265937649
        61 0.99904391031188045397420593007730566087
        62 0.99979948155661636906716009313812999866
       200 0.99998765612212040379361805662023274661
       417 0.99999997455044116794629686048124083679
      2687 0.99999999999810547903298251324014875470
   1536009 0.99999999999999044600166587966774547198
  57157270 0.99999999999999422252795052032516595669
  81436635 0.99999999999999964054666338696864815400
 143005063 0.99999999999999999572901646060974247156
 706132910 0.99999999999999999879205185535546721950
2016118929 0.99999999999999999996673765988843841836

Cf. A046959, A046964, A046965.

max = -2; aa = {}; Do[kk = N[Sin[Sqrt[n]], 50]; If[kk > max, max = kk; Print[{n, kk}]; AppendTo[aa, n]], {n, 0, 150000000}]; aa

a(14)-a(15) from David Consiglio, Jr., Jul 14 2025

cp's OEIS Frontend

A068028 Decimal expansion of 22/7.

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A068079 Decimal expansion of 355 / 113.

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A068089 Decimal expansion of 104348 / 33215.

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A046959 Numbers k where sin(k) increases monotonically to 1 (or cosec(k) decreases).

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A046964 Sin(n) decreases monotonically to -1.

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A337371 Integers k with abs(sin(k)) < 1/k.

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A172451 a(1) = 1, and for each n >= 2, a(n) is the smallest number such that 1/sin(a(n)) < 1/sin(a(k)) for all k < n, so that 1/sin(a(1)) > 1/sin(a(2)) > ... > 1/sin(a(n)) > ...

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