A225569 Decimal expansion of Sum_{n>=0} 1/10^(3^n), a transcendental number.

1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 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, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Jean-François Alcover, Jul 29 2013

Keywords

• nonn
• easy
• cons

Comments

According to the Thue-Siegel-Roth theorem, this number is transcendental.

As a sequence, characteristic sequence for powers of 3. - Franklin T. Adams-Watters, Aug 07 2013

Actually, characteristic function for 3^k - 1 (A024023), with the current starting offset 0. - Antti Karttunen, Nov 19 2017

Examples

			0.101000001000000000000000001000000000000000000000000000000000000000000000000000001...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 171.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• Wikipedia, Thue-Siegel-Roth theorem.
• Index entries for characteristic functions.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000244, A053735, A024023, A036987, A063524, A154271.

Programs

Mathematica

(* n = 4 is sufficient to get 100 digits *) Sum[1/10^(3^n), {n, 0, 4}] // RealDigits[#, 10, 100]& // First

PARI

a(n) = if(n+1 == 3^valuation(n+1, 3), 1, 0); \\ Amiram Eldar, Nov 02 2023

Formula

From Antti Karttunen, Nov 19 2017: (Start)

a(n) = A063524(A053735(1+n)).

a(n) = abs(A154271(1+n)). (End)

From Amiram Eldar, Nov 02 2023: (Start)

With offset 1:

Completely multiplicative with a(3^e) = 1, and a(p^e) = 0 for p != 3.

Dirichlet g.f.: 1/(1-3^(-s)). (End)