A021018 -id:A021018 - 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

A242827 Primes formed by the initial digits of the decimal expansion of 1/14, starting at the first nonzero digit in the expansion.

Original entry on oeis.org

7, 71, 71428571, 7142857142857, 7142857142857142857142857142857

Offset: 1

Felix Fröhlich, May 23 2014

a(6) has 104 digits. - Michel Marcus, May 26 2014

Cf. A021018.

Corresponding sequences for 1/k: A242824 (k=7), A093676 (k=12), A242826 (k=13), A242828 (k=17), A242833 (k=19).

lista(nn) = {v = [7,1,4,2,8,5]; n = 0; for (i=0, nn, n = 10*n+ v[(i % 6)+1]; if (ispseudoprime(n), print1(n, ", ")););} \\ Michel Marcus, May 26 2014

A216606 Decimal expansion of 360/7.

Original entry on oeis.org

5, 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

Offset: 2

Paul Curtz, Sep 10 2012

A020806 preceded by a 5.

Number of degrees in the exterior angle of an equilateral heptagon. Since 1969, used in many (orbiform or Reuleaux) heptagonal coins. Zambia has a natural heptagonal coin. Brazil and Costa Rica have a coin with the natural heptagon inscribed in the coin's disk.

			51.42857...

World of Coins, An Alphabet of Heptagons: Seven-sided Coins.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A019695, A020806, A021018, A060708, A068028, A178817, A178818.

Digits:=100: evalf(360/7); # Wesley Ivan Hurt, Jun 28 2016

Flatten[RealDigits[360/7, 10, 100]] (* Wesley Ivan Hurt, Jun 28 2016 *)

360/7. \\ Charles R Greathouse IV, Sep 12 2012

a(n) = 50 + 10*A020806(n).
After 5, of period 6: repeat [1, 4, 2, 8, 5, 7].
From Wesley Ivan Hurt, Jun 28 2016: (Start)
G.f.: x^3*(5-4*x+3*x^2+3*x^3+2*x^4) / (1-x+x^3-x^4).
a(n) = 9/2 + 11*cos(n*Pi)/6 + 5*cos(n*Pi/3)/3 + sqrt(3)*sin(n*Pi/3), n>2.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>6, a(n) = a(n-6) for n>8. (End)

A355068 Square array read by upwards antidiagonals: T(n,k) = k-th digit after the decimal point in decimal expansion of 1/n, for n >= 1 and k >= 1.

Original entry on oeis.org

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

Offset: 1

Chittaranjan Pardeshi, Jun 17 2022

First row is all zeros since n=1 has all zeros after the decimal point.

			Array begins:
      k=1  2  3  4  5  6  7  8
  n=1:  0, 0, 0, 0, 0, 0, 0, 0,
  n=2:  5, 0, 0, 0, 0, 0, 0, 0,
  n=3:  3, 3, 3, 3, 3, 3, 3, 3,
  n=4:  2, 5, 0, 0, 0, 0, 0, 0,
  n=5:  2, 0, 0, 0, 0, 0, 0, 0,
  n=6:  1, 6, 6, 6, 6, 6, 6, 6,
  n=7:  1, 4, 2, 8, 5, 7, 1, 4,
  n=8:  1, 2, 5, 0, 0, 0, 0, 0,
Row n=7 is 1/7 = .142857142857..., whose digits after the decimal point are 1,4,2,8,5,7,1,4,2,8,5,7, ...

Cf. A020806, A021017, A021018.
Cf. A048962, A257927.
Cf. A061480 (diagonal).
Cf. A355202 (binary).

T(n,k) = my(r=lift(Mod(10,n)^(k-1))); floor(10*r/n)%10;

def T(n,k): return (10*pow(10,k-1,n)//n)%10

1/n = Sum_{k>=1} T(n, k)*10^-k, for n > 1.

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A068028 Decimal expansion of 22/7.

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A242827 Primes formed by the initial digits of the decimal expansion of 1/14, starting at the first nonzero digit in the expansion.

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A216606 Decimal expansion of 360/7.

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A355068 Square array read by upwards antidiagonals: T(n,k) = k-th digit after the decimal point in decimal expansion of 1/n, for n >= 1 and k >= 1.

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