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.

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

1,2

Comments

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

References

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 203, Exercise 15.

Links

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

Formula

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

Mathematica

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

Prog

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

Crossrefs

Absolute values give A001511.

Indices of negative terms are listed in A091067. - M. F. Hasler, Aug 06 2015

Cf. A034947.

Sequence in context: A187808 A317673 A317954 * A001511 A265331 A347705

Adjacent sequences: A164674 A164675 A164676 * A164678 A164679 A164680

Keyword

easy,sign,mult

Author

Arie Bos, Aug 20 2009

Extensions

More terms from Alois P. Heinz, Jan 30 2012

Status

approved