A021085 -id:A021085 - cp's OEIS frontend

A021733 Decimal expansion of 1/729.

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

Offset: 0

N. J. A. Sloane

729 = 3^6 = 9^3 = 27^2.
Period is 81 = 9^2 (see example for all 81 digits of the repeating part).
Repeating part in the form of 9 X 9 square table:
  1, 3, 7, 1, 7, 4, 2, 1, 1,
  2, 4, 8, 2, 8, 5, 3, 2, 2,
  3, 5, 9, 3, 9, 6, 4, 3, 3,
  4, 7, 0, 5, 0, 7, 5, 4, 4,
  5, 8, 1, 6, 1, 8, 6, 5, 5,
  6, 9, 2, 7, 2, 9, 7, 6, 6,
  8, 0, 3, 8, 4, 0, 8, 7, 7,
  9, 1, 4, 9, 5, 1, 9, 8, 9,
  0, 2, 6, 0, 6, 3, 1, 0, 0.
Note that each column consists of 9 consecutive (cyclically repeated) digits out of 10. The missing digits in columns from left to right are {7, 6, 5, 4, 3, 2, 0, 9, 8}, which form also a cycle of 9 out of 10 consecutive digits in reverse order, all digits except 1. - Alexander Adamchuk, Dec 28 2013

			1/729 = 0.00137174211248285322359396433470507544581618655692729766\
803840877914951989026063100 (period 81). - _Alexander Adamchuk_, Dec 28 2013

Cf. A068542 (period of the fraction 1/3^n).

Cf. A010701 (1/3), A000012 (1/9), A021031 (1/27), A021085 (1/81).

RealDigits[1/729, 10, 100] (* Alexander Adamchuk, Dec 28 2013 *)

PARI
```
1/729. \\ Michel Marcus, Oct 28 2019
```

Equals Sum_{k>=1} (k*(k+1)/2)/10^(k+2). - Davide Rotondo, Jun 11 2025

A113694 Decimal expansion of 10/44955.

Original entry on oeis.org

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

Offset: 0

Daisuke Minematsu and Ryohei Miyadera, Jan 17 2006

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, 0, 0, 0, 1).

Cf. A021895, A021085.

Join[{0,0,0},RealDigits[10/44955,10,120][[1]]] (* Harvey P. Dale, May 13 2012 *)

A021247 Decimal expansion of 1/243.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 11 1996

Period 27 Repeat: [0, 0, 4, 1, 1, 5, 2, 2, 6, 3, 3, 7, 4, 4, 8, 5, 5, 9, 6, 7, 0, 7, 8, 1, 8, 9, 3]. - Wesley Ivan Hurt, May 25 2014

			0.00411522633744855967078189300411522633744855967078189300411522633744855967078...

Eric Weisstein's World of Mathematics, 243.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A010701 (1/3), A000012 (1/3^2), A021085 (1/3^4), A021733 (1/3^6).

Cf. A068542 (period of the fraction 1/3^n).

Digits:=100; evalf(1/243); # Wesley Ivan Hurt, May 25 2014

RealDigits[1/243, 10, 100, -1][[1]] (* Wesley Ivan Hurt, May 25 2014; corrected by Harvey P. Dale, Jan 23 2019 *)
PadRight[{},120,{0,0,4,1,1,5,2,2,6,3,3,7,4,4,8,5,5,9,6,7,0,7,8,1,8,9,3}] (* Harvey P. Dale, Jan 23 2019 *)

A021247_upto(N=100)={localprec(N+3);digits((1/3^5+1)\.1^N)[^1]} \\ M. F. Hasler, Apr 23 2021

1/243 = 1/3^5. - M. F. Hasler, Apr 23 2021

A190352 The continued fraction expansion of tanh(Pi) requires the computation of the pairs (p_n, q_n); sequence gives values of q_n.

Original entry on oeis.org

1, 1, 268, 1073, 15290, 16363, 48016, 64379, 176774, 417927, 594701, 1607329, 5416688, 44940833, 140239187, 185180020, 1066139287, 4449737168, 5515876455, 81672007538, 822235951835, 903907959373, 18900395139295, 719118923252583, 738019318391878

Offset: 0

N. J. A. Sloane, May 09 2011

nonn

a(2) = 268 explains the comment in A021085 that "The decimal expansion of Sum_{n>=1} floor(n * tanh(Pi))/10^n is the same as that of 1/81 for the first 268 decimal places [Borwein et al.]".

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 13.

G. C. Greubel, Table of n, a(n) for n = 0..1920

Cf. A060402, A021085.

lim:=50: with(numtheory): cfr := cfrac(tanh(Pi),lim+10,'quotients'): q[0]:=1:q[1]:=cfr[2]: printf("%d, %d, ", q[0], q[1]): for n from 2 to lim do q[n]:=cfr[n+1]*q[n-1]+q[n-2]: printf("%d, ",q[n]): od: # Nathaniel Johnston, May 10 2011

a[0] := 1; a[1] := 1; A060402:= ContinuedFraction[Tanh[Pi], 100];
a[n_]:= a[n] = A060402[[n + 1]]*a[n - 1] + a[n - 2]; Join[{1, 1}, Table[a[n], {n, 2, 75}]] (* G. C. Greubel, Apr 05 2018 *)

a(n) = A060402(n)*a(n-1) + a(n-2) for n >= 2. - Nathaniel Johnston, May 10 2011

a(4)-a(24) from Nathaniel Johnston, May 10 2011

A343614 Decimal expansion of P_{3,2}(4) = Sum 1/p^4 over primes == 2 (mod 3).

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Apr 22 2021

The prime zeta modulo function P_{m,r}(s) = Sum_{primes p == r (mod m)} 1/p^s generalizes the prime zeta function P(s) = Sum_{primes p} 1/p^s.

			P_{3,2}(4) = 0.06418614569655777899009908658740273681...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
OEIS index to entries related to the (prime) zeta function.

Cf. A003627 (primes 3k-1), A085964 (PrimeZeta(4)), A021085 (1/3^4).

Cf. A343624 (same for primes 3k+1), A086034 (for primes 4k+1), A085993 (for primes 4k+3), A343612 - A343619 (P_{3,2}(2..9): same for 1/p^2, ..., 1/p^9).

s=0;forprimestep(p=2,1e8,3,s+=1./p^4);s \\ For illustration: using primes up to 10^N gives about 3N+2 (= 26 for N=8) correct digits.

A343614_upto(N=100)={localprec(N+5); digits((PrimeZeta32(4)+1)\.1^N)[^1]} \\ see for the function PrimeZeta32.

P_{3,2}(4) = P(4) - 1/3^4 - P_{3,1}(4) = A085964 - A021085 - A343624.

A172525 a(n) = 111111111 * n.

Original entry on oeis.org

111111111, 222222222, 333333333, 444444444, 555555555, 666666666, 777777777, 888888888, 999999999, 1111111110, 1222222221, 1333333332, 1444444443, 1555555554, 1666666665, 1777777776, 1888888887, 1999999998, 2111111109, 2222222220, 2333333331, 2444444442, 2555555553

Offset: 1

Vincenzo Librandi, Feb 06 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008591, A021085.

[9*n*12345679: n in [1..20]]; // Vincenzo Librandi, Aug 20 2014

Table[9 n 12345679, {n, 1, 20}] (* Vincenzo Librandi, Aug 20 2014 *)
LinearRecurrence[{2,-1},{111111111,222222222},20] (* Harvey P. Dale, Sep 10 2023 *)

Vec(111111111*x/(x-1)^2 + O(x^100)) \\ Colin Barker, Aug 20 2014

From Colin Barker, Aug 20 2014: (Start)
a(n) = 2*a(n-1) - a(n-2).
G.f.: 111111111*x/(x-1)^2. (End)
From Elmo R. Oliveira, Jun 25 2025: (Start)
E.g.f.: 111111111*x*exp(x).
a(n) = 12345679*A008591(n). (End)

A384627 Decimal expansion of 1/998001.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jun 05 2025

The decimals of this constant contain every 3-digit number, from 000 to 999, in order, except for 998.

Periodic with period 2997.

			0.0000010020030040050060070080090100110120130140150160170...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Brady Haran and James Grime, 998,001 and its Mysterious Recurring Decimals, YouTube Numberphile video, 2012.

Cf. A021085, A034948.

Mathematica
```
First[RealDigits[1/999^2, 10, 100, -1]]
```

Equals 1/(999^2).

A021985 Decimal expansion of 1/981.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Generalization:
  1/81 = Sum_{i >= 0} 19^i/100^(i+1),
  1/981 = Sum_{i >= 0} 19^i/1000^(i+1), (this sequence)
  1/9981 = Sum_{i >= 0} 19^i/10000^(i+1). - Daniel Forgues, Oct 28 2011

Cf. A021085 (1/81).

A113657 Decimal expansion of 1/1089.

Original entry on oeis.org

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

Offset: 0

Ryohei Miyadera, Jan 16 2006

This sequence can also be produced by Sqrt[11111111111111111111111111111111111111111111] =11*Sqrt[91827364554637281910009182736455463728191], where Sqrt is the square root. In fact we found this from the square root and later looked for the same sequence in the expansion.
Comment from Eric Desbiaux, Apr 08 2008: Also, of course, decimal expansion of 9/9801. Note that
99/9801 = 0.0101010101010101010101...,
9999/9801 = 1.02020202020202020202020...,
999999/9801 = 102.0303030303030303030303...,
99999999/9801 = 10203.040404040404040404040404..., etc.

			0.0009182736455463728191000918273645546372819100091827364554...

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, 1).

Cf. A137301-A137304.

Cf. A021895, A021085.

m = 21; Sqrt[Apply[Plus, Table[11*100^k, {k, 0, m}]]]

Edited by N. J. A. Sloane, May 15 2008, at the suggestion of R. J. Mathar.

A113818 Decimal expansion of the integer (101101101101101101101101101)/9.

Original entry on oeis.org

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

Offset: 26

Ryohei Miyadera and Daisuke Minematsu, Jan 23 2006

Using square roots and periodic numbers you can produce this kind of curious sequence.

			(101101101101101101101101101)/9 = 11233455677900122344566789.

R. Miyadera, Interesting patterns made by periodic numbers and square root.

Cf. A021085, A021895, A113694, A113675.

RealDigits[101101101101101101101101101/9,10,26][[1]] (* Harvey P. Dale, May 21 2020 *)

(101101101101101101101101101)/9 or Sqrt[101101101101101101101101101], where sqrt is the square root.

Edited by N. J. A. Sloane, May 26 2006

Previous Mathematica program replaced by Harvey P. Dale, May 21 2020

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A021733 Decimal expansion of 1/729.

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A113694 Decimal expansion of 10/44955.

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A021247 Decimal expansion of 1/243.

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A190352 The continued fraction expansion of tanh(Pi) requires the computation of the pairs (p_n, q_n); sequence gives values of q_n.

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A343614 Decimal expansion of P_{3,2}(4) = Sum 1/p^4 over primes == 2 (mod 3).

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A172525 a(n) = 111111111 * n.

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A384627 Decimal expansion of 1/998001.

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