A160386 -id:A160386 - cp's OEIS frontend

A030300 Runs have lengths 2^n, n >= 0.

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

Offset: 1

Jean-Paul Delahaye

An example of a sequence with property that the fraction of 1's in the first n terms does not converge to a limit. - N. J. A. Sloane, Sep 24 2007
Image, under the coding sending a,d,e -> 1 and b,c -> 0, of the fixed point, starting with a, of the morphism a -> ab, b -> cd, c -> ee, d -> eb, e -> cc. - Jeffrey Shallit, May 14 2016
This sequence taken as digits of a base-b fraction is g(1/b) = Sum_{n>=1} a(n)/b^n = b/(b-1) * Sum_{k>=0} (-1)^k/b^(2^k) per the generating function below.  With initial 0, it is binary expansion .01001111 = A275975.  With initial 0 and digits 2*a(n), it is ternary expansion .02002222 = A160386.  These and in general g(1/b) for any integer b>=2 are among forms which Kempner showed are transcendental. - Kevin Ryde, Sep 07 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.
Kevin Ryde, Plot of A079947(n)/n, illustrating proportion of 1s in the first n terms here does not converge (but oscillates with rises and falls by hyperbolas)
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Index entries for sequences related to binary expansion of n
Index entries for characteristic functions

Cf. A030301. Partial sums give A079947.
Cf. A065359, A083905.
Characteristic function of A053738.

f0 := n->[seq(0,i=1..2^n)]; f1 := n->[seq(1,i=1..2^n)]; s := []; for i from 0 to 4 do s := [op(s), op(f1(2*i)), op(f0(2*i+1))]; od: A030300 := s;

nMax = 6; Table[1 - Mod[n, 2], {n, 0, nMax}, {2^n}] // Flatten (* Jean-François Alcover, Oct 20 2016 *)

a(n) = if(n, !(logint(n,2)%2)); /* Kevin Ryde, Aug 02 2019 */

def A030300(n): return n.bit_length()&1 # Chai Wah Wu, Jan 30 2023

a(n) = A065359(n) + A083905(n).
a(n) = (1/2)*(1+(-1)^floor(log_2(n))). - Benoit Cloitre, Feb 22 2003
G.f.: 1/(1-x) * Sum_{k>=0} (-1)^k*x^2^k. - Ralf Stephan, Jul 12 2003
a(n) = 1 - a(floor(n/2)). - Vladeta Jovovic, Aug 04 2003
a(n) = A115253(2n, n) mod 2. - Paul Barry, Jan 18 2006
a(n) = 1 - A030301(n). - Antti Karttunen, Oct 10 2017

A006467 Continued fraction for Sum_{n>=0} (-1)^n/3^(2^n).

Original entry on oeis.org

0, 4, 3, 1, 3, 5, 1, 3, 5, 3, 3, 1, 5, 3, 1, 3, 3, 5, 3, 1, 3, 5, 1, 3, 3, 5, 3, 1, 5, 3, 1, 3, 5, 3, 3, 1, 3, 5, 1, 3, 5, 3, 3, 1, 5, 3, 1, 3, 5, 3, 3, 1, 3, 5, 1, 3, 3, 5, 3, 1, 5, 3, 1, 3, 3, 5, 3, 1, 3, 5, 1, 3, 5, 3, 3, 1, 5, 3, 1, 3, 3, 5, 3, 1, 3, 5, 1, 3, 3, 5, 3, 1, 5, 3, 1, 3, 3, 5, 3, 1, 3, 5, 1, 3, 5

Offset: 0

N. J. A. Sloane

			0.234415508674864614413415474... = 0 + 1/(4 + 1/(3 + 1/(1 + 1/(3 + ...)))). - _Harry J. Smith_, May 12 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..20000
J. Shallit, Letter to N. J. A. Sloane with attachment, Aug. 1979
Jeffrey Shallit, Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), no. 2, 209-217 [DOI]

Cf. A160386 (decimal expansion). - Harry J. Smith, May 12 2009

u := 3: v := 7: Buv := [u,1,[0,u+1,u-1]]: for k from 2 to v do n := nops(Buv[3]): Buv := [u,Buv[2]+1,[seq(Buv[3][i],i=1..n-1),Buv[3][n]-(-1)^Buv[2],Buv[3][n]+(-1)^Buv[2],seq(Buv[3][n-i],i=1..n-2)]] od:seq(Buv[3][i],i=1..2^v); # first 2^v terms of A006467 # Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Dec 02 2002

{ allocatemem(932245000); default(realprecision, 20000); x=suminf(n=0, (-1)^n/3^(2^n)); x=contfrac(x); for (n=1, 20001, write("b006467.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 12 2009

Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 19 2001

A275975 Decimal expansion of Sum_{k>=0}((-1)^k/2^(2^k)).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Aug 15 2016

Except for the alternating signs, this constant is defined in a similar way to the Kempner-Mahler number A007404. It is related to the Jeffreys binary sequence A275973 somewhat like Kempner-Mahler number is related to the Fredholm-Rueppel sequence A036987.
Conjecture: Numbers of the type Sum_{k>=0}(x^(2^k)) with algebraic x and |x|<1 are known to be transcendental (Mahler 1930, Adamczewski 2013). It is likely that the alternating sign does not invalidate this property.
Yes, this number is transcendental.  It is among various such forms Kempner showed are transcendental. - Kevin Ryde, Jul 12 2019

			0.308609008556231856400340479718025221697433904166441366801367221...

Boris Adamczewski, The Many Faces of the Kempner Number, arXiv:1303.1685 [math.NT], 2013.
Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society 17 (1916), pp. 476-482.
Kurt Mahler, Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen, Math. Z. 32 (1930), 545-585.
Index entries for transcendental numbers.

Cf. A030300 (binary expansion), A160386.

Cf. A007404, A036987, A275973.

RealDigits[N[Sum[((-1)^k/2^(2^k)), {k, 0, Infinity}], 120]][[1]] (* Amiram Eldar, Jun 11 2023 *)

default(realprecision,2100);suminf(k=0,(-1)^k*0.5^2^k)

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A030300 Runs have lengths 2^n, n >= 0.

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A006467 Continued fraction for Sum_{n>=0} (-1)^n/3^(2^n).

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A275975 Decimal expansion of Sum_{k>=0}((-1)^k/2^(2^k)).

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