A034947 -id:A034947 - cp's OEIS frontend

A014710 The regular paper-folding (or dragon curve) sequence. Alphabet {2,1}.

2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2

Offset: 0

N. J. A. Sloane

nonn

Ivan Panchenko, Table of n, a(n) for n = 0..10000
G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
Index entries for sequences obtained by enumerating foldings

See A014577 for more references and more terms.

The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - N. J. A. Sloane, Jul 27 2012

Nest[Append[#1, If[EvenQ[#2], 2 - BitAnd[1, #2/2], #1[[Ceiling[#2/2]]]]] & @@ {#, Length@ #} &, {2}, 105] (* Michael De Vlieger, Apr 08 2021 *)

a(n)=if(n%2==0, 2-bitand(1,n\2), a(n\2) );
for(n=0,122,print1(a(n),", "))

def A014710(n):
    s = bin(n+1)[2:]
    m = len(s)
    i = s[::-1].find('1')
    return 2-int(s[m-i-2]) if m-i-2 >= 0 else 2 # Chai Wah Wu, Apr 08 2021

Set a=2, b=1, S(0)=a, S(n+1) = S(n)aF(S(n)), where F(x) reverses x and then interchanges a and b; sequence is limit S(infinity).

a(4*n) = 2, a(4*n+2) = 1, a(2*n+1) = a(n).

A082410 a(1)=0. Thereafter, the sequence is constructed using the rule: for any k >= 0, if a(1), a(2), ..., a(2^k+1) are known, the next 2^k terms are given as follows: a(2^k+1+i) = 1 - a(2^k+1-i) for 1 <= i <= 2^k.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Apr 24 2003

nonn

a(n) is A014577 shifted right twice (the definition here is similar to one of the constructions for A034947). - N. J. A. Sloane, Jul 27 2012
Complement of characteristic function of A060833.
From Tanya Khovanova, Apr 21 2020: (Start)
Suppose you have a deck of cards face down with 2^n cards such that the color pattern corresponds to this sequence: 0 for one color, 1 for the other. Then you proceed in the following manner: transfer to top card to the bottom of the deck, deal the next card, then repeat. The dealt cards will have alternating colors.
Even terms of this sequence alternate: 1, 0, 1, 0 and so on.
Removing even-indexed terms doesn't change the sequence. (End)

			First 3 terms are 0,1,1; therefore, a(4) = a(3+1) = 1 - a(3-1) = 1 - a(2) = 0, a(5) = a(3+2) = 1 - a(3-2) = 1 - a(1) = 1 and the sequence begins 0, 1, 1, 0, 1, ...

Index entries for sequences obtained by enumerating foldings

The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - N. J. A. Sloane, Jul 27 2012

def A082410(n):
    if n == 1:
        return 0
    s = bin(n-1)[2:]
    m = len(s)
    i = s[::-1].find('1')
    return 1-int(s[m-i-2]) if m-i-2 >= 0 else 1 # Chai Wah Wu, Apr 08 2021

For n >= 2, Sum_{k=1..n} a(k) = (n + A037834(n-1))/2.

a(1) = 0, a(4*n+2) = 1, a(4*n+4) = 0, a(2*n+1) = a(n+1) for n >= 0. - A.H.M. Smeets, Jul 27 2018

A173318 Partial sums of A005811.

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 11, 12, 14, 17, 21, 24, 26, 29, 31, 32, 34, 37, 41, 44, 48, 53, 57, 60, 62, 65, 69, 72, 74, 77, 79, 80, 82, 85, 89, 92, 96, 101, 105, 108, 112, 117, 123, 128, 132, 137, 141, 144, 146, 149, 153, 156, 160, 165, 169, 172, 174, 177, 181, 184, 186, 189

Offset: 0

Jonathan Vos Post, Feb 16 2010

Partial sums of number of runs in binary expansion of n (n>0). Partial sums of number of 1's in Gray code for n. The subsequence of squares in this partial sum begins 0, 1, 4, 9, 144, 169, 256, 289, 324 (since we also have 32 and 128 I wonder about why so many powers). The subsequence of primes in this partial sum begins: 3, 11, 17, 29, 31, 37, 41, 53, 79, 89, 101, 137, 149, 181, 191, 197, 229, 271.

Note: A227744 now gives the squares, which occur at positions given by A227743. - Antti Karttunen, Jul 27 2013

			1 has 1 run in its binary representation "1".
2 has 2 runs in its binary representation "10".
3 has 1 run in its binary representation "11".
4 has 2 runs in its binary representation "100".
5 has 3 runs in its binary representation "101".
Thus a(1) = 1, a(2) = 1+2 = 3, a(3) = 1+2+1 = 4, a(4) = 1+2+1+2 = 6, a(5) = 1+2+1+2+3 = 9.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Richard Blecksmith and Purushottam W. Laud, Some Exact Number Theory Computations via Probability Mechanisms, American Mathematical Monthly, volume 102, number 10, December 1995, pages 893-903, with a(n) = Sum_{j=0..n} b_j as calculated in section 2.
Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, volume 13, number 4, December 2017, article number 47, pages 1-43. Also first authors' copy, 2016. See example 5.5.
Kevin Ryde, Iterations of the Dragon Curve, see index "DirCumul".

Cf. A005811, A000788, A056539, A014707, A014577, A082410, A000975, A034947.
Cf. also A227737, A227741, A227742.
Cf. A227744 (squares occurring), A227743 (indices of squares).

Accumulate[Join[{0},Table[Length[Split[IntegerDigits[n,2]]],{n,110}]]] (* Harvey P. Dale, Jul 29 2013 *)

a(n) = my(v=binary(n+1),d=0,e=4); for(i=1,#v, if(v[i], v[i]=#v-i+d;d+=e;e=0, e=4)); fromdigits(v,2)>>1; \\ Kevin Ryde, Aug 27 2021

a(n) = sum(i=0..n) A005811(i) = sum(i=0..n) (A037834(i)+1) = sum(i=0..n) (A069010(i) + A033264(i)).
a(A000225(n)) = A001787(n) = A000788(A000225(n)). - Antti Karttunen, Jul 27 2013 & Aug 09 2013
a(2n) = 2*a(n) + n - 2*(ceiling(A005811(n)/2) - (n mod 2)), a(2n+1) = 2*a(n) + n + 1. - Ralf Stephan, Aug 11 2013

A209615 Completely multiplicative with a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e otherwise.

Original entry on oeis.org

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: 1

Michael Somos, Mar 10 2012

Turn sequence of the alternate paperfolding curve. Davis and Knuth define the alternate paperfolding curve by folding a long strip of paper repeatedly in half alternately to the left side or right side, then unfolding it so each crease is 90 degrees (or other angle). a(n) is their d(n) at equation 4.2. Their equation 6.2 (varied to d(2) = -1 as described there) is equivalent to the definition here. The curve is drawn by a unit step forward, turn a(1)*90 degrees left, a unit step forward, turn a(2)*90 degrees left, and so on. - Kevin Ryde, Apr 18 2020

			G.f. = x - x^2 - x^3 + x^4 + x^5 + x^6 - x^7 - x^8 + x^9 - x^10 - x^11 - x^12 + ...
From _Kevin Ryde_, Apr 18 2020: (Start)
                   ...              alternate
                    | -1           paperfolding
            -1 --->\ \<--- +1         curve
             ^   -1 |      ^
             |      v      |      turns +1 left
  start --> +1     +1 ---> +1        or -1 right
(End)

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.

Jianing Song, Table of n, a(n) for n = 1..10000
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]
Kevin Ryde, Iterations of the Alternate Paperfolding Curve, section "Turn".
Index to divisibility sequences

Cf. A002144, A003324, A034947, A106665, A292077.
Indices of 1: A338692 = A016813 U A343501; indices of -1: A338691 = A004767 U A343500.
Inverse Moebius transform gives A338690.

A209615[n_] := JacobiSymbol[-1, n]*(-1)^IntegerExponent[n, 2];
Array[A209615, 100] (* Paolo Xausa, Feb 26 2025 *)

{a(n) = my(v); if( n==0, 0, v = valuation( n, 2); (-1)^(n/2^v\2 + v))};

{a(n) = if( n!=0, -kronecker( -1, n) * (-1)^if( n!=0, 1 - valuation( n, 2) %2))};

{a(n) = my(A, p, e, f); sign(n) * if( n==0, 0, A = factor(abs(n)); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; (-1)^(e * (p%4 != 1))) )};

def A209615(n): return -1 if ((n>>(m:=(~n&n-1).bit_length()))+1>>1)+m&1^1 else 1 # Chai Wah Wu, Feb 25 2025

G.f.: Sum_{k>=0} (-1)^k * x^(2^k) / (1 + x^(2^(k+1))).
G.f. A(x) satisfies A(x) + A(x^2) = x / (1 + x^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (v + w) - (u + v)^2 * (1 + 2*(v + w)).
If p is prime then a(p) = 1 if and only if p is in A002144.
a(4*n + 1) = 1, a(4*n + 3) = -1. a(2*n) = a(3*n) = a(-n) = -a(n).
a(n) = -(-1)^A106665(n-1) unless n=0.
a(2n) = -a(n), a(2n+1) = (-1)^n.  [Davis and Knuth equation 4.2] - Kevin Ryde, Apr 18 2020
From Jianing Song, Apr 24 2021: (Start)
a(n) = 1 <=> A003324(n) = 1 or 4, a(n) = -1 <=> A003324(n) = 2 or 3. In other words, a(n) = Legendre(A003324(n), 5) == A003324(n)^2 (mod 5).
a(n) = A034947(n) * (-1)^(v2(n)), where v2(n) = A007814(n) is the 2-adic valuation of n.
Dirichlet g.f.: beta(s)/(1 + 2^(-s)). (End)

A014709 The regular paper-folding (or dragon curve) sequence. Alphabet {1,2}.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1

Offset: 0

N. J. A. Sloane

nonn

Over the alphabet {a,b} this is aabaabbaaabbabbaaabaabbbaabbabbaaaba...

With offset 1, completely multiplicative modulo 3. - Peter Munn, Jun 20 2022

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 155, 182.
G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.

Ivan Panchenko, Table of n, a(n) for n = 0..10000
Gabriele Fici and Luca Q. Zamboni, On the least number of palindromes contained in an infinite word, Theoretical Computer Science, Volume 481, 2013, pp. 1-8. See page 1.
Index entries for sequences obtained by enumerating foldings

See A014577 for more references and more terms.
The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - N. J. A. Sloane, Jul 27 2012
Cf. A065339.

(3 - JacobiSymbol[-1, Range[100]])/2 (* Paolo Xausa, May 26 2024 *)

a(n)=if(n%2==0, 1+bitand(1,n\2), a(n\2) );
for(n=0,122,print1(a(n),", "))

Set a=1, b=2, S(0)=a, S(n+1) = S(n)aF(S(n)), where F(x) reverses x and then interchanges a and b; sequence is limit S(infinity).
a(4n) = 1, a(4n+2) = 2, a(2n+1) = a(n).
a(n) = (3-jacobi(-1,n+1))/2 (cf. A034947). - N. J. A. Sloane, Jul 27 2012 [index adjusted by Peter Munn, Jun 22 2022]
a(n) = 1 + A065339(n+1) mod 2. - Peter Munn, Jun 20 2022

A302033 a(n) = A019565(A003188(n)).

Original entry on oeis.org

1, 2, 6, 3, 15, 30, 10, 5, 35, 70, 210, 105, 21, 42, 14, 7, 77, 154, 462, 231, 1155, 2310, 770, 385, 55, 110, 330, 165, 33, 66, 22, 11, 143, 286, 858, 429, 2145, 4290, 1430, 715, 5005, 10010, 30030, 15015, 3003, 6006, 2002, 1001, 91, 182, 546, 273, 1365, 2730, 910, 455, 65, 130, 390, 195, 39, 78, 26, 13, 221, 442, 1326, 663, 3315, 6630, 2210, 1105

Offset: 0

Antti Karttunen & Peter Munn, Apr 16 2018

A squarefree analog of A207901 (and the subsequence consisting of its squarefree terms): Each term is either a divisor or a multiple of the next one, and the terms differ by a single prime factor. Compare also to A284003.

For all n >= 0, max(a(n + 1), a(n)) / min(a(n + 1), a(n)) = A094290(n + 1) = prime(valuation(n + 1, 2) + 1) = A000040(A001511(n + 1)). [See Russ Cox's Dec 04 2010 comment in A007814.] - David A. Corneth & Antti Karttunen, Apr 16 2018

Antti Karttunen, Table of n, a(n) for n = 0..8191

A permutation of A005117. Subsequence of A207901.
Cf. A302054 (gives the sum of prime divisors).
Cf. A000040, A001511, A003188, A019565, A034947, A059897, A064706, A094290.
Cf. also A277811, A283475, A284003.

Array[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[BitXor[#, Floor[#/2]], 2] &, 72, 0] (* Michael De Vlieger, Apr 27 2018 *)

A003188(n) = bitxor(n, n>>1);
A019565(n) = {my(j); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A302033(n) = A019565(A003188(n));

first(n) = {my(pr = primes(1 + logint(n, 2)), ex = vector(#pr, i, 1), res = vector(n)); res[1] = 1; for(i = 1, n-1, v = valuation(i, 2); res[i + 1] = res[i] * pr[v++] ^ ex[v]; ex[v]*=-1); res}

a(n) = A019565(A003188(n)).
a(n) = A284003(A064706(n)).
a(n+1) = A059897(a(n), A094290(n+1)). - Peter Munn, Apr 01 2019

A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 0, 4, 1, 4, 0, 0, 2, 0, 0, 5, 2, 2, 0, 6, 0, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 6, 0, 4, 0, 3, 2, 0, 0, 8, 2, 0, 0, 0, 2, 0, 0, 0, 1, 6, 0, 6, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 7, 4, 0, 0, 6, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 0, 10, 1, 4, 0, 0, 4, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 2, 2, 0, 9, 2, 0, 0, 8, 0

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 3*x^4 + 2*x^5 + 4*x^8 + x^9 + 4*x^10 + 2*x^13 + 5*x^16 + 2*x^17 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002175, A008441, A019669, A053692, A113407, A121444.
Inverse Moebius transform of A034947.
Sum_{d|n} Kronecker(k, d): A035143..A035181 (k=-47..-9, skipping numbers that are not cubefree), A035182 (k=-7), A192013 (k=-6), A035183 (k=-5), A002654 (k=-4), A002324 (k=-3), A002325 (k=-2), this sequence (k=-1), A000012 (k=0), A000005 (k=1), A035185 (k=2), A035186 (k=3), A001227 (k=4), A035187..A035229 (k=5..47, skipping numbers that are not cubefree).

a[n_] := DivisorSum[n, KroneckerSymbol[-1, #] &]; Array[a, 105] (* Jean-François Alcover, Dec 02 2015 *)

{a(n) = if( n<1, 0, direuler( p=2, n, 1/((1 - X) * (1 - kronecker( -1, p) * X))) [n])}; /* Michael Somos, Jan 05 2012 */

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -1, d)))}; /* Michael Somos, Jan 05 2012 */

a(n) is multiplicative with a(2^e) = e + 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4). - Michael Somos, Jan 05 2012
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n).
Dirichlet g.f.: zeta(s)*beta(s)/(1 - 2^(-s)), where beta is the Dirichlet beta function. - Ralf Stephan, Mar 27 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 = 1.570796... (A019669). - Amiram Eldar, Oct 17 2022

A106665 Alternate paper-folding (or alternate dragon curve) sequence.

Original entry on oeis.org

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

Offset: 0

Duane K. Allen (computeruser(AT)sprintmail.com), May 13 2005

Regular dragon curve (A014577) sequence results from repeated folding of long strip of paper in half in the same direction, say right to left. This alternate dragon curve sequence results from repeated folding of long strip of paper in half in alternating directions, right to left, then left to right and so forth.

In the Wikipedia article "Dragon Curve" note the illustrated description under the heading "[Un]Folding the Dragon" and note that the 1's and 0's in the A106665 description correspond to the L and R folds in the Wikipedia discussion. - Robert Munafo, Jun 03 2010

			1 + x^3 + x^4 + x^5 + x^8 + x^12 + x^13 + x^15 + x^16 + x^19 + x^20 + ...

M. Gardner, "The Dragon Curve and Other Problems (Mathematical Games)", Scientific American, 1967, columns for March, April, July.
M. Gardner, "Mathematical Magic Show" (contains the dragon curve columns).
D. E. Knuth, "Art of Computer Programming," vol. 2, 3rd. ed., "Seminumerical > Algorithms," (section 4.1)

Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.3.2 "The alternate paper-folding sequence", pp. 90-92
Chandler Davis and Donald E. Knuth, Number Representations and Dragon Curves - II, Journal of Recreational Mathematics, Vol. 3, No. 3, 1970, pp. 133-149, see section 4. Reprinted in Donald E. Knuth, Selected Papers on Fun and Games, 2011, pages 571-614.
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]
William J. Gilbert, Fractal geometry derived from complex bases, The Mathematical Intelligencer, Volume 4, Number 2 (June 1982), pp. 78-86. (ISSN 0343-6993; DOI 10.1007/BF03023486)
Kevin Ryde, Iterations of the Alternate Paperfolding Curve, see index "TurnLpred" with a(n) = TurnLpred(n+1).
Wikipedia, Dragon curve

Cf. A014577, A209615.

Cf. A034947, A096268.

a:= proc(n) option remember;
      `if`(irem(n, 4)=0, 1, `if`(irem(n, 2)=1, 1-a((n-1)/2), 0))
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, Mar 10 2012

a[n_] := a[n] = Switch[Mod[n, 4], 0, 1, 2, 0, _, 1-a[(n-1)/2]];
Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Mar 15 2017 *)

{a(n) = n++; if( n==0, 0, v = valuation( n, 2); (n/2^v\2 + v+1) %2 )} /* Michael Somos, Mar 10 2012 */

def A106665(n): return ((n+1>>(m:=(~(n+1)&n).bit_length()))+1>>1)+m&1 # Chai Wah Wu, Feb 25 2025

For n >= 0, a(4n) = 1, a(4n+2) = 0, a(2n+1) = 1 - a(n).
(-1)^a(n) = -A034947(n+1) * (-1)^A096268(n). - Alec Edgington (alec(AT)obtext.com), Aug 02 2010
-(-1)^a(n) = A209615(n+1). - Michael Somos, Mar 10 2012
G.f.: 1/(1 - x^4) + Sum_{k>=1} x^(2*((-1 - (-2)^(k - 1)) mod 2^(k + 1)) + 1)/(1 - x^(2^(k + 2))). - Miles Wilson, Nov 18 2024

Edited by N. J. A. Sloane, Jun 04 2010 to include material from discussions on the Sequence Fans Mailing List.

A164677 For a binary reflected Gray code, the (Hamming/Euclidean) distance between 2 subsequent points x and y is 1, say in coordinate k. If y has a 1 in coordinate k and x has a 0, than (x,y) is indicated by k, if it is the other way around, (x,y) is indicated by -k. The sequence has a fractal character such that G(d+1) = G(d) d+1 R(G(d)) where R(G(d)) alters d --> -d and leaves all other numbers invariant.

Original entry on oeis.org

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

Offset: 1

Arie Bos, Aug 20 2009

This is the paper-folding sequence Fold(1,2,3,4,5,...). It is also the fixed point of the map 1->1,2; 2->-1,3; 3->-1,4; 4->-1,5; ...; -1->1,-2; -2->-1,-3; -3->-1,-4; -4->-1,-5; ... [Allouche and Shallit]. - N. J. A. Sloane, Jul 27 2012

Multiplicative because both A034947 and A001511 are. - Andrew Howroyd, Aug 06 2018

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 203, Exercise 15.

Alois P. Heinz, Table of n, a(n) for n = 1..8192
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See pp. 2, 26.
Index entries for sequences obtained by enumerating foldings.

Absolute values give A001511.
Indices of negative terms are listed in A091067. - M. F. Hasler, Aug 06 2015
Cf. A034947.

a[n_] := KroneckerSymbol[-1, n] * IntegerExponent[2n, 2];
Array[a, 80] (* Jean-François Alcover, Sep 08 2019 *)

A164677(n)=(valuation(n,2)+1)*if(n>>valuation(n,2)%4==3,-1,1) \\ M. F. Hasler, Aug 06 2015

a(n) = (-1)^chi_A091067(n)*A001511(n), where chi_A091067 is the characteristic function of A091067. - M. F. Hasler, Aug 06 2015
a(n) = A034947(n)*A001511(n). - Andrew Howroyd, Aug 06 2018
Multiplicative with a(2^e) = e+1, and a(p^e) = (-1)^(e*(p-1)/2) for an odd prime p. - Amiram Eldar, Jun 09 2025

More terms from Alois P. Heinz, Jan 30 2012

A226162 a(n) = Kronecker Symbol (-5/n), n >= 0.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, May 29 2013

The number of -1's among the four terms following the 0 at a(5*k), for k >= 0, is 1, 2, 3, 3, 1, 0, 3, 2, 2, 1, 4, 4, 0, 1, 3, 3, 1, 1, 3, 4, ...
See the Weisstein link, where it is stated that the period length is 0.
In general, the sequence {(k/n)} is not periodic if and only if k == 3 (mod 4). - Jianing Song, Dec 30 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Leo Goldmakher, Mock characters and the Kronecker symbol, arXiv:1608.03957 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Kronecker Symbol (contains this sequence).
Wikipedia, Kronecker Symbol

Cf. A035183 (inverse Moebius transform).
Sequences related to Kronecker symbols that do not form a Dirichlet character: this sequence {(-5/n)}, A034947 {(-1/n)}, A091338 {(3/n)}, A089509 {(7/n)}.
Cf. A080891 (5/n), A100047.

0, seq(numtheory:-jacobi(-5, n), n=1..89); # Peter Luschny, Dec 30 2018

Mathematica
```
Table[KroneckerSymbol[-5, n],{n,0,89}]
```

a(n)=kronecker(-5,n); \\ Andrew Howroyd, Jul 23 2018

Completely multiplicative with a(2) = -1, a(5) = 0, a(p) = 1 if p == 1, 3, 7, 9 (mod 20), a(p) = -1 if p == 11, 13, 17, 19 (mod 20). - Jianing Song, Dec 30 2018

Keyword:mult added by Andrew Howroyd, Jul 23 2018

cp's OEIS Frontend

A014710 The regular paper-folding (or dragon curve) sequence. Alphabet {2,1}.

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A082410 a(1)=0. Thereafter, the sequence is constructed using the rule: for any k >= 0, if a(1), a(2), ..., a(2^k+1) are known, the next 2^k terms are given as follows: a(2^k+1+i) = 1 - a(2^k+1-i) for 1 <= i <= 2^k.

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A173318 Partial sums of A005811.

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A209615 Completely multiplicative with a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e otherwise.

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A014709 The regular paper-folding (or dragon curve) sequence. Alphabet {1,2}.

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A302033 a(n) = A019565(A003188(n)).

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A035184 a(n) = Sum_{d|n} Kronecker(-1, d).

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