A164677 - OEIS

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

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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

Jean-Paul Allouche and Jeffrey 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.

Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and Ziggu Puzzles, LIPIcs 366, 13th Int'l Conf. Fun Algor. (FUN 2026), Art. 22. See p. 2.

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

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

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.