A030300 -id:A030300 - cp's OEIS frontend

A079947 Partial sums of A030300.

1, 1, 1, 2, 3, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36

Offset: 1

N. J. A. Sloane, Feb 22 2003

nonn

Kevin Ryde, Table of n, a(n) for n = 1..8192
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 49-50.

Accumulate@ Flatten@ Table[1 - Mod[n, 2], {n, 0, 6}, {2^n}] (* Michael De Vlieger, Oct 29 2022 *)

a(n) = my(k=logint(n,2), p=(2<Kevin Ryde, Jul 13 2019 */

def A079947(n): return n-((1<Chai Wah Wu, Jan 30 2023

a(n) = (n+1+(2/3)*(4^e_4-1)+(-1)^e_2*(n-1-2*(4^e_4-1)))/2 where e_4=floor(log[4](n)) and e_2=floor(log[2](n))=floor(log[4](n^2)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 22 2003

a(n) = n - A079954(n). Let k=A000523(n), then a(n) = n-A000975(k) if k even, or a(n) = A000975(k) if k odd. - Kevin Ryde, Jul 13 2019

A071859 Determinant of M_n where M_n is the n X n matrix m_(i,j) = A030300(i+j).

Original entry on oeis.org

0, 0, -1, 1, 1, -1, 0, 0, 1, 1, -1, -1, 0, 0, -1, 1, 1, 1, -2, 2, 3, -3, -1, 1, -2, -2, 1, 1, 1, -1, 0, 0, 1, 1, -1, 1, 2, -2, -1, -1, 3, 3, -2, -2, -1, 1, -1, -1, 0, 0, 1, -1, -1, 1, 0, 0, 1, 1, -1, -1, 0, 0, -1, 1, 1, 1, -2, 2, 3, -3, -1, -1, 4, 4, -3, -3

Offset: 1

Benoit Cloitre, Jun 09 2002

a(n)=0 for n=1,2,7,8,13,14,31,32,49,50,55,56,61,62...hence if a(n)=0, n odd it seems that a(n+1)=0. It seems also that if a(n)=0 (n odd ) then n == 1 mod 6 (converse doesn't hold).

Cf. A030300.

for(n=1,70,print1(matdet(matrix(n,n,i,j,length(binary(j+i))%2)),","))

Name corrected by Nathaniel Johnston, Sep 26 2011

A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

Original entry on oeis.org

1, -1, 0, -2, 1, -1, -1, -3, 2, 0, 0, -2, 0, -2, -2, -4, 3, 1, 1, -1, 1, -1, -1, -3, 1, -1, -1, -3, -1, -3, -3, -5, 4, 2, 2, 0, 2, 0, 0, -2, 2, 0, 0, -2, 0, -2, -2, -4, 2, 0, 0, -2, 0, -2, -2, -4, 0, -2, -2, -4, -2, -4, -4, -6, 5, 3, 3, 1, 3, 1, 1, -1, 3

Offset: 0

Clark Kimberling

-Sum_{n>=1} a(n)/((2*n)*(2*n+1)) = the "alternating Euler constant" log(4/Pi) = 0.24156... - (see A094640 and Sondow 2005, 2010).

a(A072600(n)) < 0; a(A072601(n)) <= 0; a(A031443(n)) = 0; a(A072602(n)) >= 0; a(A072603(n)) > 0; a(A031444(n)) = 1; a(A031448(n)) = -1; abs(a(A089648(n))) <= 1. - Reinhard Zumkeller, Feb 07 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.
Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).
Index entries for sequences related to binary expansion of n

Cf. A031443 for n when a(n)=0, A053738 for n when a(n) odd, A053754 for n when a(n) even, A030300 for a(n+1) mod 2.
Cf. A066879, A094640, A110625, A145037.
Cf. A072600, A072601, A072602, A072603, A031444, A031448, A089648.
See A268289 for a recurrence based on this sequence.

a037861 n = a023416 n - a000120 n  -- Reinhard Zumkeller, Aug 01 2013

A037861:= proc(n) local L;
     L:= convert(n,base,2);
     numboccur(0,L) - numboccur(1,L)
end proc:
map(A037861, [$0..100]); # Robert Israel, Mar 08 2016

Table[Count[ IntegerDigits[n, 2], 0] - Count[IntegerDigits[n, 2], 1], {n, 0, 75}]

a(n) = if (n==0, 1, 1 + logint(n, 2) - 2*hammingweight(n)); \\ Michel Marcus, May 15 2020 and Jun 16 2020

def A037861(n):
    return 2*format(n,'b').count('0')-len(format(n,'b')) # Chai Wah Wu, Mar 07 2016

From Henry Bottomley, Oct 27 2000: (Start)
a(n) = A023416(n) - A000120(n) = A029837(n) - 2*A000120(n) = 2*A023416(n) - A029837(n).
a(2*n) = a(n) + 1; a(2*n + 1) = a(2*n) - 2 = a(n) - 1. (End)
G.f. satisfies A(x) = (1 + x)*A(x^2) - x*(2 + x)/(1 + x). - Franklin T. Adams-Watters, Dec 26 2006
a(n) = b(n) for n > 0 with b(0) = 0 and b(n) = b(floor(n/2)) + (-1)^(n mod 2). - Reinhard Zumkeller, Dec 31 2007
G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(2^k)*(x^(2^k) - 1)/(1 + x^(2^k)). - Ilya Gutkovskiy, Apr 07 2018

A065359 Alternating bit sum for n: replace 2^k with (-1)^k in binary expansion of n.

Original entry on oeis.org

0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2, -1, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2, -1, 0, 1, -1, 0, -2, -1, -3, -2, -1, 0, -2, -1, 0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2, -1, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 0, 1, 2, 3, 1, 2, 0, 1, -1, 0, 1, 2, 0, 1, -1, 0, -2

Offset: 0

Marc LeBrun, Oct 31 2001

Notation: (2)[n](-1)
From David W. Wilson and Ralf Stephan, Jan 09 2007: (Start)
a(n) is even iff n in A001969; a(n) is odd iff n in A000069.
a(n) == 0 (mod 3) iff n == 0 (mod 3).
a(n) == 0 (mod 6) iff (n == 0 (mod 3) and n/3 not in A036556).
a(n) == 3 (mod 6) iff (n == 0 (mod 3) and n/3 in A036556). (End)
a(n) = A030300(n) - A083905(n). - Ralf Stephan, Jul 12 2003
From Robert G. Wilson v, Feb 15 2011: (Start)
First occurrence of k and -k: 0, 1, 2, 5, 10, 21, 42, 85, ..., (A000975); i.e., first 0 occurs for 0, first 1 occurs for 1, first -1 occurs at 2, first 2 occurs for 5, etc.;
  a(n)=-3 only if n mod 3 = 0,
  a(n)=-2 only if n mod 3 = 1,
  a(n)=-1 only if n mod 3 = 2,
  a(n)= 0 only if n mod 3 = 0,
  a(n)= 1 only if n mod 3 = 1,
  a(n)= 2 only if n mod 3 = 2,
  a(n)= 3 only if n mod 3 = 0, ..., . (End)
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - Philippe Deléham, Oct 20 2011
In the Koch curve, number the segments starting with n=0 for the first segment. The net direction (i.e., the sum of the preceding turns) of segment n is a(n)*60 degrees. This is since in the curve each base-4 digit 0,1,2,3 of n is a sub-curve directed respectively 0, +60, -60, 0 degrees, which is the net 0,+1,-1,0 of two bits in the sum here. - Kevin Ryde, Jan 24 2020

			Alternating bit sum for 11 = 1011 in binary is 1 - 1 + 0 - 1 = -1, so a(11) = -1.

Antti Karttunen, Table of n, a(n) for n = 0..65535 (terms 0..1000 from Harry J. Smith)
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197. [Preprint.]
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
H.-K. Hwang, S. Janson and T.-H. Tsai. Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications. Preprint, 2016.
H.-K. Hwang, S. Janson and T.-H. Tsai. Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications. ACM Transactions on Algorithms, 13:4 (2017), #47. DOI:10.1145/3127585.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
Helge von Koch, Une Méthode Géométrique Élémentaire pour l'Étude de Certaines Questions de la Théorie des Courbes Planes, Acta Arithmetica, volume 30, 1906, pages 145-174.
William Paulsen, wpaulsen(AT)csm.astate.edu, Partitioning the [prime] maze
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A005536 (partial sums), A056832 (abs first differences), A010060 (mod 2), A039004 (indices of 0's).
Cf. A000120, A065360, A065081, A019565, A195017.
Cf. also A004718.
Cf. analogous sequences for bases 3-10: A065368, A346688, A346689, A346690, A346691, A346731, A346732, A055017 and also A373605 (for primorial base).

a065359 0 = 0
a065359 n = - a065359 n' + m where (n', m) = divMod n 2
-- Reinhard Zumkeller, Mar 20 2015

A065359 := proc(n) local dgs ; dgs := convert(n,base,2) ; add( -op(i,dgs)*(-1)^i,i=1..nops(dgs)) ; end proc: # R. J. Mathar, Feb 04 2011

f[0]=0; f[n_] := Plus @@ (-(-1)^Range[ Floor[ Log2@ n + 1]] Reverse@ IntegerDigits[n, 2]); Array[ f, 107, 0]

a(n) = my(s=0, u=1); for(k=0,#binary(n)-1,s+=bittest(n,k)*u;u=-u);s /* Washington Bomfim, Jan 18 2011 */

a(n) = my(b=binary(n)); b*[(-1)^k|k<-[-#b+1..0]]~; \\ Ruud H.G. van Tol, Oct 16 2023

a(n) = if(n==0, 0, 2*hammingweight(bitand(n, ((4<<(2*logint(n,4)))-1)/3)) - hammingweight(n)) \\ Andrew Howroyd, Dec 14 2024

def a(n):
    return sum((-1)**k for k, bi in enumerate(bin(n)[2:][::-1]) if bi=='1')
print([a(n) for n in range(107)]) # Michael S. Branicky, Jul 13 2021

from sympy.ntheory import digits
def A065359(n): return sum((0,1,-1,0)[i] for i in digits(n,4)[1:]) # Chai Wah Wu, Jul 19 2024

G.f.: (1/(1-x)) * Sum_{k>=0} (-1)^k*x^2^k/(1+x^2^k). - Ralf Stephan, Mar 07 2003
a(0) = 0, a(2n) = -a(n), a(2n+1) = 1-a(n). - Ralf Stephan, Mar 07 2003
a(n) = Sum_{k>=0} A030308(n,k)*(-1)^k. - Philippe Deléham, Oct 20 2011
a(n) = -a(floor(n/2)) + n mod 2. - Reinhard Zumkeller, Mar 20 2015
a(n) = A139351(n) - A139352(n). - Kevin Ryde, Jan 24 2020
G.f. A(x) satisfies: A(x) = x / (1 - x^2) - (1 + x) * A(x^2). - Ilya Gutkovskiy, Jul 28 2021
a(n) = A195017(A019565(n)). - Antti Karttunen, Jun 19 2024

More terms from Ralf Stephan, Jul 12 2003

A059448 The parity of the number of zero digits when n is written in binary.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Feb 02 2001

Old name was: "If A_k are the terms from 2^(k-1) through to 2^k-1, then A_(k+1) is B_k A_k where B_k is A_k with 0's and 1's swapped, starting from a(1)=0; also parity of number of zero digits when n is written in binary. a(0) not given as it could be 1 or 0 depending on the definition or formula used." - Michel Dekking, Sep 11 2020
The sequence (when prefixed by 0) is overlap-free [Allouche and Shallit].
From Vladimir Shevelev, May 23 2017: (Start)
Theorem: The sequence is cubefree.
Here we show only that the sequence contains no three consecutive equal terms. Indeed, using the recursions below, we have
a(4*n)=a(n), a(4*n+1)=1-a(n), a(4*n+2)=1-a(n), a(4*n+3)=a(n), n >= 1, and our statement easily follows. In general, the Theorem could be proved either directly (cf. A269027) or using the remark below from Jeffrey Shallit and the well-known fact [first proved not later than 1912 by Axel Thue (private communication from Jean-Paul Allouche)] that the Thue-Morse sequence is cubefree.
Note that, by the formulas modulo 4, the sequence is constructed over four terms {a(4*n),a(4*n+1),a(4*n+2),a(4*n+3)} which, starting with a(4), are either {0,1,1,0} or {1,0,0,1}, the first elements of which form {a(n)}. (End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 26, Problem 23.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jeffrey Shallit, Arseny M. Shur, and Stefan Zorcic, New constructions for 3-free and 3+-free binary morphisms, arXiv:2310.15064 [math.CO], 2023. Mentions this sequence.
Index entries for sequences related to binary expansion of n
Index entries for characteristic functions

Characteristic function of A059009.
Cf. A298952 (complement), A242179 (values +-1).
Cf. A023416 (A080791), A054429, A010059, A010060, A106400.

a059448 = (`mod` 2) . a023416  -- Reinhard Zumkeller, Mar 01 2012

s1:=[];
for n from 1 to 200 do
t1:=convert(n,base,2); t2:=subs(1=NULL,t1); s1:=[op(s1),nops(t2) mod 2]; od:
s1;

Table[Boole[OddQ[Count[IntegerDigits[n, 2], 0]]], {n, 1, 105}] (* Jean-François Alcover, Apr 05 2013 *)

a(n)=(#binary(n)-hammingweight(n))%2;
vector(99,n,a(n)) /* Joerg Arndt, Sep 11 2020 */

def A059448(n): return (n.bit_length()^n.bit_count())&1 # Chai Wah Wu, Jul 26 2023

a(2n) = 1 - a(n); a(2n+1) = a(n) = 1 - a(2n). If 2^k <= n < 2^(k+1) then a(n) = 1 - a(n-2^(k-1)). a(n) = A023416(n) mod 2 = A059009(n) - 2n = 2n + 1 - A059010(n) = |A010060(n) - A030300(n-1)|.
Let b(1)=1 and b(n) = b(n-ceiling(n/2)) - b(n-floor(n/2)); then for n >= 1, a(n) = (1/2)*(1-b(2n+1)). - Benoit Cloitre, Apr 26 2005
Alternatively, if x is the sequence, then x = 010 mu^2(x), where mu is the Thue-Morse morphism sending 0 to 01 and 1 to 10. - Jeffrey Shallit, Jun 06 2016
a(n) = A010059(A054429(n)) = (1+A008836(A163511(n)))/2. - Antti Karttunen, May 30 2017
Alternatively, if x is the sequence, then x = 0 tau(x), where tau is the "twisted" Thue-Morse morphism sending 0 to 10 and 1 to 01. Note that tau^2 = mu^2, giving x = 010 mu^2(x). - Michel Dekking, Sep 30 2020

Name changed by Michel Dekking, Sep 11 2020

A147991 Sequence S such that 1 is in S and if x is in S, then 3x-1 and 3x+1 are in S.

Original entry on oeis.org

1, 2, 4, 5, 7, 11, 13, 14, 16, 20, 22, 32, 34, 38, 40, 41, 43, 47, 49, 59, 61, 65, 67, 95, 97, 101, 103, 113, 115, 119, 121, 122, 124, 128, 130, 140, 142, 146, 148, 176, 178, 182, 184, 194, 196, 200, 202, 284, 286, 290, 292, 302, 304, 308, 310, 338, 340, 344, 346

Offset: 1

Clark Kimberling, Dec 07 2008

nonn

Positive numbers that can be written in balanced ternary without a 0 trit. - J. Hufford, Jun 30 2015

Let S be the set of terms. Define c: Z -> P(R) so that c(m) is the translated Cantor ternary set spanning [m-0.5, m+0.5], and let C be the union of c(m) for all m in S U {0} U -S. C is the closure of the translated Cantor ternary set spanning [-0.5, 0.5] under multiplication by 3. - Peter Munn, Jan 31 2022

			0th generation: 1;
1st generation: 2 4;
2nd generation: 5 7 11 13.

Robert Israel, Table of n, a(n) for n = 1..10000
Gevorg Hmayakyan, Trig identity for a(n)
J. H. Loxton and A. J. van der Poorten, An Awful Problem About Integers in Base Four, Acta Arithmetica, volume 49, 1987, pages 193-203. In section 7, John Selfridge and Carole Lacampagne ask whether every k != 0 (mod 3) is the quotient of two terms of this sequence (cf. A351243 and A006288).
Eric Weisstein's World of Mathematics, Cantor Set
Eric Weisstein's World of Mathematics, Closure

Cf. A168542, A191108, A306556.
Cf. A006288, A351243 (non-quotients).
See also the related sequences listed in A191106.
One half of each position > 0 where A307744 sets or equals a record.
Cf. A030300.
Column k=3 of A360099.

import Data.Set (singleton, insert, deleteFindMin)
a147991 n = a147991_list !! (n-1)
a147991_list = f $ singleton 1 where
   f s = m : (f $ insert (3*m - 1) $ insert (3*m + 1) s')
         where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 21 2012, Jan 23 2011

A147991:= proc(n) option remember; if n::even then 3*procname(n/2)-1 else 3*procname((n-1)/2)+1 fi end proc:
A147991(1):= 1:
[seq](A147991(i),i=1..1000); # Robert Israel, May 05 2014

nn=346; s={1}; While[s1=Select[Union[s, 3*s-1, 3*s+1], # <= nn &];  s != s1, s=s1]; s
a[ n_] := If[ n < -1 || n > 0, 3 a[Quotient[n, 2]] - (-1)^Mod[n, 2], 0]; (* Michael Somos, Dec 22 2018 *)

{a(n) = if( n<-1 || n>0, 3*a(n\2) - (-1)^(n%2), 0)}; /* Michael Somos, Dec 22 2018 */

a(n) = fromdigits(apply(b->if(b,1,-1),binary(n)), 3); \\ Kevin Ryde, Feb 06 2022

a(n) = 3*a(n/2) - 1 if n>=2 is even, 3*a((n-1)/2) + 1 if n is odd, a(0)=0. - Robert Israel, May 05 2014
G.f. g(x) satisfies g(x) = 3*(x+1)*g(x^2) + x/(1+x). - Robert Israel, May 05 2014
Product_{j=0..n-1} cos(3^j) = 2^(-n+1)*Sum_{i=2^(n-1)..2^n-1} cos(a(i)). - Gevorg Hmayakyan, Jan 15 2017
Sum_{i=2^(n-1)..2^n-1} cos(a(i)/3^(n-1)*Pi/2) = 0. - Gevorg Hmayakyan, Jan 15 2017
a(n) = -a(-1-n) for all n in Z. - Michael Somos, Dec 22 2018
For n > 0, A307744(2*a(2n)) = A307744(2*a(2n+1)) = A307744(2*a(n)) + 1. - Peter Munn, Jan 31 2022
a(n) mod 2 = A030300(n). - Alois P. Heinz, Jan 29 2023

A360099 To get A(n,k), replace 0's in the binary expansion of n with (-1) and interpret the result in base k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, -1, 0, 1, 0, 1, 0, 1, 1, 2, -1, 0, 1, 2, 3, -1, 1, 0, 1, 3, 4, 1, 1, -1, 0, 1, 4, 5, 5, 3, 1, 1, 0, 1, 5, 6, 11, 7, 5, 3, -1, 0, 1, 6, 7, 19, 13, 11, 7, -2, 1, 0, 1, 7, 8, 29, 21, 19, 13, 1, 0, -1, 0, 1, 8, 9, 41, 31, 29, 21, 14, 3, 0, 1, 0, 1, 9, 10, 55, 43, 41, 31, 43, 16, 5, 2, -1

Offset: 0

Alois P. Heinz, Jan 25 2023

The empty bit string is used as binary expansion of 0, so A(0,k) = 0.

			Square array A(n,k) begins:
   0,  0, 0,  0,  0,   0,   0,   0,   0,   0,   0, ...
   1,  1, 1,  1,  1,   1,   1,   1,   1,   1,   1, ...
  -1,  0, 1,  2,  3,   4,   5,   6,   7,   8,   9, ...
   1,  2, 3,  4,  5,   6,   7,   8,   9,  10,  11, ...
  -1, -1, 1,  5, 11,  19,  29,  41,  55,  71,  89, ...
   1,  1, 3,  7, 13,  21,  31,  43,  57,  73,  91, ...
  -1,  1, 5, 11, 19,  29,  41,  55,  71,  89, 109, ...
   1,  3, 7, 13, 21,  31,  43,  57,  73,  91, 111, ...
  -1, -2, 1, 14, 43,  94, 173, 286, 439, 638, 889, ...
   1,  0, 3, 16, 45,  96, 175, 288, 441, 640, 891, ...
  -1,  0, 5, 20, 51, 104, 185, 300, 455, 656, 909, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-6, 10 give: A062157, A145037, A006257, A147991, A147992, A153777, A147993, A359925.
Rows n=0-10 give: A000004, A000012, A023443, A000027(k+1), A165900, A002061, A165900(k+1), A002061(k+1), A083074, A152618, A062158.
Main diagonal gives A360096.
Cf. A030300, A057427.

A:= proc(n, k) option remember; local m;
     `if`(n=0, 0, k*A(iquo(n, 2, 'm'), k)+2*m-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= (n, k)-> (l-> add((2*l[i]-1)*k^(i-1), i=1..nops(l)))(Bits[Split](n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

G.f. for column k satisfies g_k(x) = k*(x+1)*g_k(x^2) + x/(1+x).
A(n,k) = k*A(floor(n/2),k)+2*(n mod 2)-1 for n>0, A(0,k)=0.
A(n,k) mod 2 = A057427(n) if k is even.
A(n,k) mod 2 = A030300(n) if k is odd and n>=1.
A(2^(n+1),1) + n = 0.

A030301 n-th run has length 2^(n-1).

Original entry on oeis.org

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

Offset: 1

Jean-Paul Delahaye

Cf. A000523, A029837.
Cf. A030300. Partial sums give A079954.
Characteristic function of A053754 (after its initial 0).

[Floor(Log(n)/Log(2)) mod 2: n in [1..100]]; // Vincenzo Librandi, Jun 23 2015

nMax = 7; Table[1 - Mod[n, 2], {n, nMax}, {2^(n-1)}] // Flatten (* Jean-François Alcover, Oct 20 2016 *)
Table[{PadRight[{},2^(n-1),0],PadRight[{},2^n,1]},{n,1,8,2}]//Flatten (* Harvey P. Dale, Apr 12 2023 *)

PARI
```
a(n)=if(n<1,0,1-length(binary(n))%2)
```

a(n)=if(n<1,0,if(n%2==0,-a(n/2)+1,-a((n-1)/2)+1-(((n-1)/2)==0))) /* Ralf Stephan */

def A030301(n): return n.bit_length()&1^1 # Chai Wah Wu, Jan 30 2023

a(n) = A000523(n) mod 2 = (A029837(n+1)+1) mod 2.
a(n) = 0 iff n has an odd number of digits in binary, = 1 otherwise. - Henry Bottomley, Apr 06 2000
a(n) = (1/2)*{1-(-1)^floor(log(n)/log(2))}. - Benoit Cloitre, Nov 22 2001
a(n) = 1-a(floor(n/2)). - Vladeta Jovovic, Aug 04 2003
a(n) = 1 - A030300(n). - Antti Karttunen, Oct 10 2017

A160386 Decimal expansion of Sum_{n>=0}(-1)^n/3^(2^n).

Original entry on oeis.org

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

Offset: 0

Harry J. Smith, May 12 2009

This sum is among forms which Kempner showed are transcendental. - Kevin Ryde, Sep 16 2019

			0.234415508674864614413415474345604597614724563906693027679427737098381...

Harry J. Smith, Table of n, a(n) for n = 0..20000
Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society, Vol. 17, No. 4 (October 1916), pp. 476-482.
Index entries for transcendental numbers.

Cf. A006467 (continued fraction), A030300 (ternary expansion).

RealDigits[N[Sum[(-1)^n/3^(2^n), {n, 0, Infinity}], 120]][[1]] (* Amiram Eldar, Jun 11 2023 *)

{ default(realprecision, 20080); x=10*suminf(n=0, (-1)^n/3^(2^n)); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b160386.txt", n, " ", d)); }

A153777 Sequence S such that 1 is in S and if x is in S, then 5x-1 and 5x+1 are in S.

Original entry on oeis.org

1, 4, 6, 19, 21, 29, 31, 94, 96, 104, 106, 144, 146, 154, 156, 469, 471, 479, 481, 519, 521, 529, 531, 719, 721, 729, 731, 769, 771, 779, 781, 2344, 2346, 2354, 2356, 2394, 2396, 2404, 2406, 2594, 2596, 2604, 2606, 2644, 2646, 2654, 2656, 3594, 3596, 3604

Offset: 1

Clark Kimberling, Jan 02 2009

nonn

Subsequences include A003463, A083065.
1st generation: 1
2nd generation: 4, 6
3rd generation: 19, 21, 29, 31
4th generation: 94, 96, 104, 106, 144, 146, 154, 156
Does every generation contain p or 2p for some prime p?

Harvey P. Dale, Table of n, a(n) for n = 1..10000
Gevorg Hmayakyan, Trig identity for a(n)

Cf. A003463, A030300, A083065, A153773, A153776.

Column k=5 of A360099.

nxt[n_]:=Flatten[5#+{1,-1}&/@n]; Union[Flatten[NestList[nxt,{1},5]]] (* Harvey P. Dale, Dec 25 2012 *)

Product_{j=0..n-1} cos(5^j) = 2^(-n+1)*Sum_{i=2^(n-1)..2^n-1} cos(a(i)). - Gevorg Hmayakyan, Jan 15 2017
Sum_{i=2^(n-1)..2^n-1} cos(a(i)/5^(n-1)*Pi/2) = 0. - Gevorg Hmayakyan, Jan 15 2017
a(n) mod 2 = A030300(n). - Alois P. Heinz, Jan 29 2023

cp's OEIS Frontend

A079947 Partial sums of A030300.

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A071859 Determinant of M_n where M_n is the n X n matrix m_(i,j) = A030300(i+j).

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A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

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A065359 Alternating bit sum for n: replace 2^k with (-1)^k in binary expansion of n.

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A059448 The parity of the number of zero digits when n is written in binary.

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A147991 Sequence S such that 1 is in S and if x is in S, then 3x-1 and 3x+1 are in S.

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