A034947 -id:A034947 - cp's OEIS frontend

A059129 A hierarchical sequence (W2{2}* - see A059126).

1, 2, 1, 2, 3, 2, 1, 2, 1, 3, 4, 3, 1, 2, 1, 2, 3, 2, 1, 2, 1, 4, 5, 4, 1, 2, 1, 2, 3, 2, 1, 2, 1, 3, 4, 3, 1, 2, 1, 2, 3, 2, 1, 2, 1, 5, 6, 5, 1, 2, 1, 2, 3, 2, 1, 2, 1, 3, 4, 3, 1, 2, 1, 2, 3, 2, 1, 2, 1, 4, 5, 4, 1, 2, 1, 2, 3, 2, 1, 2, 1, 3, 4, 3, 1, 2, 1, 2, 3, 2, 1, 2, 1, 6, 7, 6, 1, 2, 1, 2, 3, 2, 1, 2, 1

Offset: 0

Jonas Wallgren, Jan 19 2001

Begin with the empty finite sequence s_0. Inductively extend s_n to obtain s_{n+1} as follows: if s_n is given by a, b, c, ..., d, e, f, with g being the least integer that is not a value of s_n, then s_{n+1} is a, b, c, ..., d, e, f, g, -f, -e, -d, ..., -c, -d, -a, -g. The terms of {a(n)} give the absolute values of the limit of these sequences. These finite sequences naturally describe elements of fundamental groups occurring in picture-hanging puzzles and Brunnian links. - Thomas Anton, Oct 15 2022

Jonas Wallgren, Hierarchical sequences
E. D. Demaine, M. L. Demaine, Y. N. Minsky, J. S. B. Mitchell, R. L. Rivest, and M. Patrascu, Picture-Hanging Puzzles, arXiv:1203.3602, [cs.DS], 2012-2014.

Cf. A034947, A059126.

A204617 Multiplicative with a(p^e) = p^(e-1)*H(p). H(2) = 1, H(p) = p - 1 if p == 1 (mod 4) and H(p) = p + 1 if p == 3 (mod 4).

Original entry on oeis.org

1, 1, 4, 2, 4, 4, 8, 4, 12, 4, 12, 8, 12, 8, 16, 8, 16, 12, 20, 8, 32, 12, 24, 16, 20, 12, 36, 16, 28, 16, 32, 16, 48, 16, 32, 24, 36, 20, 48, 16, 40, 32, 44, 24, 48, 24, 48, 32, 56, 20, 64, 24, 52, 36, 48, 32, 80, 28, 60, 32, 60, 32, 96, 32, 48, 48, 68, 32

Offset: 1

José María Grau Ribas, Jan 17 2012

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A072437.
Cf. A175647, A243381.
Cf. A079458, A062570.

with(numtheory):
a := n->add(jacobi(-1,d)*mobius(d)*n/d, d in divisors(n)):
seq(a(n), n = 1..60); # Peter Bala, Dec 26 2023

ar[p_,s_] := Which[Mod[p,4]==1, p^(s-1)*(p-1), Mod[p,4]==3, p^(s-1)*(p+1), True,p^(s-1)]; arit[1] = 1; arit[n_] := Product[ar[FactorInteger[n][[i,1]], FactorInteger[n][[i,2]]], {i, Length[FactorInteger[n]]}]; Array[arit, 100]

A204617(n) = { my(f=factor(n),p); prod(i=1, #f~, p=f[i, 1]; (p^(f[i, 2]-1)) * if(2==p,1,if(1==(p%4),p-1,p+1))); }; \\ Antti Karttunen, Nov 16 2021

a(n) = phi(n) if n is in A072437.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (3/8) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2) * Product_{primes p == 3 (mod 4)} (1 + 1/p^2) = 3*A243381/(8*A175647) = 0.409404... . - Amiram Eldar, Dec 24 2022
a(n) = n*Product_{primes p, p | n} (1 - A034947(p)/p) = Sum_{d | n} A034947(d)* mobius(d)*n/d. Cf. A000010(n) = Sum_{d | n} mobius(d)*n/d. - Peter Bala, Dec 26 2023
a(n) = A079458(n)/A062570(n). - Ridouane Oudra, Jun 04 2024

A322016 a(0) = 0; for n > 0, if A003188(n) > A003188(n-1) then a(n) = n-1, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 1, 0, 3, 4, 0, 0, 7, 8, 9, 0, 0, 12, 0, 0, 15, 16, 17, 0, 19, 20, 0, 0, 0, 24, 25, 0, 0, 28, 0, 0, 31, 32, 33, 0, 35, 36, 0, 0, 39, 40, 41, 0, 0, 44, 0, 0, 0, 48, 49, 0, 51, 52, 0, 0, 0, 56, 57, 0, 0, 60, 0, 0, 63, 64, 65, 0, 67, 68, 0, 0, 71, 72, 73, 0, 0, 76, 0, 0, 79, 80, 81, 0, 83, 84, 0, 0, 0, 88, 89, 0, 0, 92, 0, 0, 0, 96, 97

Offset: 0

Antti Karttunen, Nov 24 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A003188, A034947, A322015, A322017.

With[{nn = 98}, {0, 0}~Join~Most@ ReplacePart[ConstantArray[0, nn], Map[# -> # &, Position[Partition[Array[BitXor[#, Floor[#/2]] &, nn], 2, 1], ?(First@ # < Last@ # &), {1}, Heads -> False][[All, 1]] ]]] (* _Michael De Vlieger, Nov 25 2018 *)
Join[{0,0},With[{t=Partition[Table[BitXor[n,Floor[n/2]],{n,100}],2,1]},Flatten[ If[#[[1]]<#[[2]],Position[t,#],0]&/@t]]] (* Harvey P. Dale, Jan 28 2021 *)

A003188(n) = bitxor(n, n>>1);
A322016(n) = if(0==n,0,if(A003188(n)>A003188(n-1),n-1,0));

A322016(n) = if(!n,0,(((kronecker(-1,n)+1)/2)*(n-1)));

a(0) = 0; for n > 0, a(n) = (1/2)*(A034947(n)+1)*(n-1).

A337821 For n >= 0, a(4n+1) = 0, a(4n+3) = a(2n+1) + 1, a(2n+2) = a(n+1).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 2, 3, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 0, 3, 4, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 2, 3, 1, 0, 0, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 0, 4, 5, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 0, 2, 3, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 1, 2, 0, 3, 4, 1

Offset: 1

Peter Munn, Sep 23 2020

This sequence is the ruler sequence A007814 interleaved with this sequence; specifically, the odd bisection is A007814, the even bisection is the sequence itself.
The 3-adic valuation of the Doudna sequence (A005940).
The 2-adic valuation of Kimberling's paraphrase (A003602) of the binary number system. [Edited Peter Munn, Aug 13 2025.]

			Start of table showing the interleaving with ruler sequence, A007814:
   n  a(n)  A007814    a(n/2)
            ((n+1)/2)
   1   0       0
   2   0                 0
   3   1       1
   4   0                 0
   5   0       0
   6   1                 1
   7   2       2
   8   0                 0
   9   0       0
  10   0                 0
  11   1       1
  12   1                 1
  13   0       0
  14   2                 2
  15   3       3
  16   0                 0
  17   0       0
  18   0                 0
  19   1       1
  20   0                 0
  21   0       0
  22   1                 1
  23   2       2
  24   1                 1

Odd bisection: A007814.
A000265, A003602, A005940, A007949 are used in a formula defining this sequence.
Positions of zeros: A091072.
Sequences with similar interleaving: A089309, A014577, A025480, A034947, A038189, A082392, A099545, A181363, A274139.

a[n_] := IntegerExponent[(n/2^IntegerExponent[n, 2] + 1)/2, 2]; Array[a, 100] (* Amiram Eldar, Sep 30 2020 *)

a(n) = valuation(n>>valuation(n,2)+1, 2) - 1; \\ Kevin Ryde, Apr 06 2024

a(2*n) = a(n).
a(2*n+1) = A007814(n+1).
a(n) = A007949(A005940(n)).
a(n) = A007814(A003602(n)) = A007814((A000265(n)+1) / 2) = A089309(n) - 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Sep 13 2024

A371594 Starting positions of runs in the paperfolding sequence A014707.

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 13, 14, 16, 19, 20, 22, 25, 27, 29, 30, 32, 35, 36, 38, 40, 43, 45, 46, 49, 51, 52, 54, 57, 59, 61, 62, 64, 67, 68, 70, 72, 75, 77, 78, 80, 83, 84, 86, 89, 91, 93, 94, 97, 99, 100, 102, 104, 107, 109, 110, 113, 115, 116, 118, 121, 123, 125

Offset: 1

Jeffrey Shallit, Mar 28 2024

A "run" is a maximal block of consecutive identical terms. The paperfolding sequence A014707 is more usually indexed starting at position 1, not 0, and this choice is reflected in the sequence (cf. A034947).

			The first few terms of A014707 are 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, and runs begin at positions 1, 3, 4, 6, 8, 11, 13, 14, ...

M. Bunder, B. Bates, and S. Arnold, The summed paperfolding sequence, Bull. Aust. Math. Soc. 110 (2024), 189-198.
Kevin Ryde, Iterations of the Dragon Curve, see index "TurnRunStart" with a(n) = TurnRunStart(n-1).
Jeffrey Shallit, Automaton for A371594.

Cf. A014707, A034947.

Abs@ SplitBy[Array[#  KroneckerSymbol[-1, #] &, 120], Sign][[All, 1]] (* Michael De Vlieger, Mar 28 2024 *)

a(n) = if(n==1,1, n--; 2*n + bitxor(bittest(n,0), bittest(n,valuation(n,2)+1))); \\ Kevin Ryde, Apr 06 2024

# DFA transition function and simulation
d = { (0,0):0, (0,1):1, (1,0):2, (1,1):3, (2,0):4, (2,1):5,
      (3,0):6, (3,1):7, (4,0):4, (4,1):5, (5,0):2, (5,1):3,
      (6,0):0, (6,1):1, (7,0):6, (7,1):7 }
def ok(n):
    q, w = 0, map(int, bin(n)[2:])
    for c in w: q = d[q, c]
    return q in {1, 3, 4, 6}
print([k for k in range(126) if ok(k)]) # Michael S. Branicky, Mar 28 2024

# using formula and function in A014707
def a(n): return 2*n-1 - (n + A014707(n-2))%2 if n>=2 else 1
print([a(n) for n in range(1, 64)]) # Michael S. Branicky, Mar 29 2024

The automaton accompanying this entry accepts exactly the base-2 representations of the terms of this sequence.

a(n) = 2*n-1 - ((n + A014707(n-2)) mod 2), for n >= 2. - Kevin Ryde, Mar 28 2024

A112347 Kronecker symbol (-1, n) except a(0) = 0.

Original entry on oeis.org

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, 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, 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, 1, 1, -1, 1, 1, -1

Offset: 0

Michael Somos, Sep 12 2005

			x + x^2 - x^3 + x^4 + x^5 - x^6 - x^7 + x^8 + x^9 + x^10 - x^11 - x^12 + x^13 + ...

J.-P. Allouche and J. Shallit, On three conjectures of P. Barry, arxiv preprint arXiv:2006.04708 [math.NT], June 8 2020.
Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
Index entries for sequences obtained by enumerating foldings

Cf. A034947.

First differences of A005811.

Join[{0},KroneckerSymbol[-1,Range[110]]] (* Harvey P. Dale, Jun 02 2019 *)

PARI
```
{a(n) = if( n, kronecker( -1, n))}
```

Multiplicative with a(2^e) = 1, a(p^e) = (-1)^(e(p-1)/2) if p>2.
a(2n) = a(n), a(4*n + 1) = 1, a(4*n + 3) = -1. a(-n) = -a(n).
a(n) = A034947(n) unless n=0.

A317627 a(n) = A317253 - floor(8*n/3).

Original entry on oeis.org

2, 3, 1, 1, 3, 1, 2, 1, -1, 1, 3, 1, 2, 3, 1, 1, 1, -1, 2, 1, -1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 2, 1, -1, 1, 1, -1, 2, 3, 1, 1, 1, -1, 2, 1, -1, 1, 3, 1, 2, 3, 1, 1, 3, 1, 2, 1, -1, 1, 3, 1, 2, 3, 1, 1, 1, -1, 2, 1, -1, 1, 1, -1, 2, 3, 1, 1, 3, 1, 2

Offset: 1

A.H.M. Smeets, Aug 02 2018

sign

Cf. A000034, A014577, A034947,A089607, A317414.

n,f,i,p,q,base = 1,1,0,0,1,3
while i < 100000:
    i,p,q = i+1,p*base,q*base
    if i == f:
        p,n = p+1,n+1
        f = f*n
n,a,j = 0,0,0
while p%q > 0:
    a,f,p,q = a+1,p//q,q,p%q
    if f == 1:
        n = n+1
        print(n,a-1-(8*n//3))

a(3*n+1) = 2 - (n mod 2) for n >= 0, a(6*n+2) = 3 - 2*(n mod 2) and a(6*n+5) = a(3*n+2) for n >= 0, a(6*n+3) = 1 - 2*(n mod 2) and a(6*n+6) = a(3*n+3) for n >= 0.
a(3*n+1) = A000034(n+1) for n >= 0.
a(3*n+2) = A089607(n) for n > 1.
a(3*n+2) = 2*A014577(n-1)+1 for n > 0.
a(3*n+3) = A034947(n) = 2*A014577(n-1)-1 for n > 0.

A143347 Decimal expansion of the paper-folding constant, or the dragon constant.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Aug 09 2008

Named "the Gaussian Liouville number" by Borwein and Coons (2008). - Amiram Eldar, Apr 29 2021

			0.85073618820186726036...

Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves -- I and II, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, CSLI Publications, 2010, pages 571-614.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 6.8.5 Paper Folding, pages 439-440.

Joerg Arndt, Matters Computational (The Fxtbook), p. 744.
Peter Borwein and Michael Coons, Transcendence of the Gaussian Liouville number and relatives, arXiv:0806.1694 [math.NT], 2008.
Michael J. Coons, Some aspects of analytic number theory: parity, transcendence, and multiplicative functions, Ph.D. Thesis, Department of Mathematics, Simon Fraser University, 2009.
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves, Journal of Recreational Mathematics, volume 3, number 2, April 1970, pages 66-81, and number 3, July 1970, pages 133-149. [Cached copy, with permission]
J. H. Loxton, A method of Mahler in transcendence theory and some of its applications, Bulletin of the Australian Mathematical Society, Vol. 29, No. 1 (1984), pp. 127-136.
Michel Mendès France and Alf van der Poorten, Arithmetic and Analytic Properties of Paper Folding Sequences, Bulletin of the Australian Mathematical Society, volume 24, issue 1, 1981, pages 123-131.
A. J. van der Poorten and J. H. Loxton, Arithmetic properties of the solutions of a class of functional equations, Journal für die reine und angewandte Mathematik, Vol. 330 (1982), pp. 159-172; alternative link.
Eric Weisstein's World of Mathematics, Paper Folding Constant.
Index entries for sequences obtained by enumerating foldings
Index entries for transcendental numbers

Cf. A014577 (binary expansion), A034947.

RealDigits[ N[ Sum[ 8^2^k/(2^2^(k + 2) - 1), {k, 0, Infinity}], 110]][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 26 2012 *)

default(realprecision,510);
c=sum(k=0, 10, 1.0/( 2^(2^k) * ( 1 - 1/(2^(2^(k+2))) ) ) )
/* Joerg Arndt, Aug 28 2011 */

Equals Sum_{k>=1} A014577(k)/2^k = Sum_{k>=1} (1+A034947(k))/2^(k+1). - Amiram Eldar, Apr 29 2021

A286299 First differences of A286298.

Original entry on oeis.org

1, 2, -1, 2, 1, -1, -1, 2, 1, 1, -1, -1, 1, -1, -1, 2, 1, 1, -1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, 2, 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, 2, 1, 1

Offset: 0

N. J. A. Sloane, Jun 02 2017

sign

A014577 is obtained if sequence terms are changed as follows: 2's replaced by 1's and -1's replaced by 0's. - Karyn McLellan, Aug 16 2018

Danielle Cox and K. McLellan, A problem on generation sets containing Fibonacci numbers, Fib. Quart., 55 (No. 2, 2017), 105-113.

Cf. A286298.

a(n) = 2*A034947(n) if n = 2^k for k>1, otherwise a(n) = A034947(n). - Karyn McLellan, Aug 16 2018

A323378 Square array read by antidiagonals: T(n,k) = Kronecker symbol (-n/k), n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 0, -1, 1, -1, 1, 1, 1, 0, 0, 0, 1, 1, -1, -1, 1, -1, -1, 1, 0, 1, 0, -1, 0, -1, 1, 1, 0, 1, 1, 0, -1, 1, 1, 0, -1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 1, -1, -1, 1, -1, -1, 1, -1, 1, 1, 1, -1

Offset: 1

Jianing Song, Jan 12 2019

If A215200 is arranged into a square array A215200(n,k) = kronecker symbol(n/k) with n >= 0, k >= 1, then this sequence gives the other half of the array.

Note that there is no such n such that the n-th row and the n-th column are the same.

			Table begins
  1,  1, -1,  1,  1, -1, -1,  1,  1,  1, ... ((-1/k) = A034947)
  1,  0,  1,  0, -1,  0, -1,  0,  1,  0, ... ((-2/k) = A188510)
  1, -1,  0,  1, -1,  0,  1, -1,  0,  1, ... ((-3/k) = A102283)
  1,  0, -1,  0,  1,  0, -1,  0,  1,  0, ... ((-4/k) = A101455)
  1, -1,  1,  1,  0, -1,  1, -1,  1,  0, ... ((-5/k) = A226162)
  1,  0,  0,  0,  1,  0,  1,  0,  0,  0, ... ((-6/k) = A109017)
  1,  1, -1,  1, -1, -1,  0,  1,  1, -1, ... ((-7/k) = A175629)
  1,  0,  1,  0, -1,  0, -1,  0,  1,  0, ... ((-8/k) = A188510)
  ...

Eric Weisstein's World of Mathematics, Kronecker Symbol.

Cf. A215200.

The first rows are listed in A034947, A188510, A102283, A101455, A226162, A109017, A175629, A188510, ...

PARI
```
T(n,k) = kronecker(-n, k)
```

cp's OEIS Frontend

A059129 A hierarchical sequence (W2{2}* - see A059126).

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A204617 Multiplicative with a(p^e) = p^(e-1)*H(p). H(2) = 1, H(p) = p - 1 if p == 1 (mod 4) and H(p) = p + 1 if p == 3 (mod 4).

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A322016 a(0) = 0; for n > 0, if A003188(n) > A003188(n-1) then a(n) = n-1, otherwise a(n) = 0.

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A337821 For n >= 0, a(4n+1) = 0, a(4n+3) = a(2n+1) + 1, a(2n+2) = a(n+1).

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A371594 Starting positions of runs in the paperfolding sequence A014707.

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A112347 Kronecker symbol (-1, n) except a(0) = 0.

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A317627 a(n) = A317253 - floor(8*n/3).

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A143347 Decimal expansion of the paper-folding constant, or the dragon constant.

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