OFFSET
0,3
COMMENTS
Continued fraction expansion of sqrt(3) - 1. - N. J. A. Sloane, Dec 17 2007. Cf. A040001, A048878/A002530.
Minimum number of terms required to express n as a sum of odd numbers.
Shadow transform of even numbers A005843. - Michel Marcus, Jun 06 2013
From Jianing Song, Nov 01 2022: (Start)
For n > 0, a(n) is the minimal gap of distinct numbers coprime to n. Proof: denote the minimal gap by b(n). For odd n we have A058026(n) > 0, hence b(n) = 1. For even n, since 1 and -1 are both coprime to n we have b(n) <= 2, and that b(n) >= 2 is obvious.
The maximal gap is given by A048669. (End)
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..20000
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Digital Root.
Eric Weisstein's World of Mathematics, Ternary.
FORMULA
a(n) = n if n <= 2, otherwise a(A053735(n)).
a(n) = 0 if n=0, otherwise A000034(n-1).
a(n) = ((n+1) mod 2) + 2*sign(n) - 1. - Wesley Ivan Hurt, Dec 06 2013
Multiplicative with a(2^e) = 2, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Aug 06 2018
a(0) = A055034(1) / A000010(1), a(n) = A000010(n+1) / A055034(n+1), n>1. - Torlach Rush, Oct 29 2019
Dirichlet g.f.: zeta(s)*(1+1/2^s). - Amiram Eldar, Jan 01 2023
EXAMPLE
0.732050807568877293527446341... = 0 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...)))). - Harry J. Smith, May 31 2009
MAPLE
MATHEMATICA
Table[Mod[n + 1, 2] + 2 Sign[n] - 1, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 06 2013 *)
PROG
(PARI) { allocatemem(932245000); default(realprecision, 12000); x=contfrac(sqrt(3)-1); for (n=0, 20000, write("b134451.txt", n, " ", x[n+1])); } [Harry J. Smith, May 31 2009]
(Haskell)
a134451 = until (< 3) a053735
-- Reinhard Zumkeller, May 12 2011
CROSSREFS
KEYWORD
nonn,base,easy,mult
AUTHOR
Reinhard Zumkeller, Oct 27 2007
STATUS
approved