A010701 Constant sequence: the all 3's sequence.

3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset

0,1

Comments

Decimal expansion of 1/3. - Raymond Wang, Mar 06 2010

Continued fraction expansion of (3+sqrt(13))/2. - Bruno Berselli, Mar 15 2011

1/3 is the asymptotic probability that the greatest common divisor of two positive integers selected independently at random is not a 5-rough number (A047229). - Amiram Eldar, Feb 24 2026

1/3 is the probability that a chord defined by two points independently and uniformly selected at random on the circumference of a circle is longer than a side of an equilateral triangle inscribed in this circle (the first solution of Bertrand's problem, 1889). - Amiram Eldar, Apr 18 2026

Also digits of the 4-adic integer -1. - Stefano Spezia, May 19 2026

References

William Dunham, Journey Through Genius, Wiley, 1990, Chapter 8, pp. 193-194.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.3.1, p. 281.

Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See p. 220.

Links

Table of n, a(n) for n=0..80.

Joseph Bertrand, Calcul des probabilités, Gauthier-Villars, Paris, 1889, pp. 5-6.

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., Vol. 6 (2003), Article 03.1.6.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1011.

Tanya Khovanova, Recursive Sequences.

Rick Mabry, Proof without words: 1/4+(1/4)^2+(1/4)^3+...=1/3, Math. Mag., Vol. 72, No. 1 (1999), p. 63.

Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.

Index entries for linear recurrences with constant coefficients, signature (1).

Index to divisibility sequences.

Formula

G.f.: 3/(1-x). - Bruno Berselli, Mar 15 2011

E.g.f.: 3*e^x. - Vincenzo Librandi, Jan 24 2012

a(n) = A040000(n) + A054977(n). - Reinhard Zumkeller, May 06 2012

a(n) = 3*A000012(n). - Michel Marcus, Dec 18 2015

a(n) = floor(1/(n - cot(1/n))). - Clark Kimberling, Mar 10 2020

Equals Sum_{k>=1} (1/4)^k (as a constant). - Michel Marcus, Jun 11 2020

Equals Sum_{k>=2} (k-1)/binomial(2*k,k) (as a constant). - Amiram Eldar, Jun 05 2021

Equals Sum_{k>=1} (-1)^(k+1)/2^k. - Michal Paulovic, Mar 02 2023

Equals Product_{k>=2} (1 - 2/(k*(k + 1))) = Product_{k>=2} (k - 1)*(k + 2)/(k*(k+1)) (as a constant) [Knopp]. - Stefano Spezia, Jan 29 2026

Example

1/3 = 0.33333333333333333333333333333333333333333333... - Bruno Berselli, Mar 21 2014

Maple

evalf(1/3, 100); # Michal Paulovic, Mar 02 2023

Mathematica

Table[3, {100}] (* Wesley Ivan Hurt, Jul 16 2014 *)

Prog

(Haskell)
a010701 = const 3
a010701_list = repeat 3  -- Reinhard Zumkeller, May 07 2012
(Maxima) makelist(3, n, 0, 30); /* Martin Ettl, Nov 09 2012 */
(PARI) a(n)=3 \\ Felix Fröhlich, Jul 16 2014
(Python)
def A010701(n): return 3 # Chai Wah Wu, Nov 10 2022

Crossrefs

Cf. A000012, A040000, A047229, A054977, A007283 (binomial transform).

Sequence in context: A179804 A368311 A102818 * A290858 A174971 A122553

Adjacent sequences: A010698 A010699 A010700 * A010702 A010703 A010704

Keyword

nonn,cons,easy

Author

N. J. A. Sloane

Status

approved