OFFSET
0,1
COMMENTS
Continued fraction expansion of 3+sqrt(10). - Bruno Berselli, Mar 15 2011
Decimal expansion of Sum_{n >= 0} n/binomial(2*n+1, n) = 2/3. - Bruno Berselli, Sep 14 2015
From Amiram Eldar, Feb 24 2026: (Start)
2/3 is the asymptotic probability that the greatest common divisor of two positive integers selected independently at random is a 5-rough number (A007310).
In general, the asymptotic probability that the greatest common divisor of two positive integers selected independently at random is a p-rough number (number that is not divisible by the primes less than p) is Product_{prime q < p} (1-1/q^2). (End)
Also digits of the 7-adic integer -1. - Stefano Spezia, May 19 2026
REFERENCES
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.
Fernando Q. Gouvêa, p-Adic Numbers: An Introduction, Springer-Verlag, 2020; see p. 16.
Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See exercise 85 at p. 228.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.
LINKS
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1014.
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (1).
FORMULA
G.f.: 6/(1-x). - Bruno Berselli, Mar 15 2011
E.g.f.: 6*e^x. - Vincenzo Librandi, Jan 27 2012
Equals Product_{k>=2} (k^3 - 1)/(k^3 + 1) (as a constant) [Knopp]. - Stefano Spezia, Jan 29 2026
MATHEMATICA
a[n_] := 6; Table[a[n], {n, 1, 100}] (* Jason Bard, Nov 25 2025 *)
PROG
(PARI) a(n)=6 \\ Charles R Greathouse IV, Sep 28 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved