A123070 Hofstadter Flip-G-sequence.

0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 40, 41, 42, 42, 43, 44

Offset: 0

Author

Albert Neumueller (albert.neu(AT)gmail.com), Sep 28 2006

Keywords

• nonn
• changed

Comments

This sequence is an answer to the question posed on page 137 of Goedel, Escher, Bach: find a sequence that generates a tree which is like that of the G-sequence (see A005206), but flipped and renumbered, so that when the nodes of the tree are read from bottom-to-top, left-to-right, the natural numbers: 1, 2, 3, 4, 5, 6, ... are obtained.

14.15.16.17.18.19.20.21

.\..\./...\./..\...\./

..9..10....11...12.13

...\./......\...\./

....6........7...8

.....\........\./

......4........5

........\..../

..........3

............\

.............2

..............\ /

...............1

To construct the tree: node n is connected to the node flip-G(n) = a(n) below it:

n

.\

.a(n)

For example:

7

.\

..5

since a(7) = 5. The tree has a recursive structure, since the following construct

\

.x

..\ /

...x

can be repeatedly added on top of its own ends, to construct the tree from its root: e.g.

\

.x

..\./...\

...x.....x

....\.....\ /

.....x.....x

......\.../

........x

References

• D. Hofstadter, "Goedel, Escher, Bach", p. 137.

Links

• David Fifield, Table of n, a(n) for n = 0..10000
• Alan Michael Gómez Calderón, A Proof for a Hofstadter Flip-G-sequence Closed-form formula
• Alan Michael Gómez Calderón, A Proof using Walnut for a Hofstadter Flip-G-sequence Closed-form formula
• Pierre Letouzey, Hofstadter's problem for curious readers, technical report, 2015.
• Physics Forum Discussion, Recursive Trees (from GED).
• Jeffrey Shallit, Fibonacci automaton for the Flip-G sequence.
• Index entries for sequences from "Goedel, Escher, Bach"
• Index entries for Hofstadter-type sequences

Crossrefs

Cf. A005206, A060144.

Programs

Mathematica

a[n_] := a[n] = Switch[n, 0, 0, 1, 1, 2, 1, 3, 2, _, (n+1) - a[a[n-1]+1]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 16 2019 *)

Python

# Emulate a breadth-first traversal of the tree defining the "flip-tree" of the Hofstadter G-sequence.
def gflip_iter():
    yield 0
    yield 1
    # Start on a left-branch node, parent node is 1.
    queue = [(1, 1)]
    n = 2
    while True:
        parent, state = queue.pop(0)
        yield parent
        if state == 0:
            # Root node. Add the two children.
            queue.append((n, 1))
            queue.append((n, 0))
        elif state == 1:
            # Left-branch node. Add a new root.
            queue.append((n, 0))
        n += 1
i = gflip_iter()
for n in range(0, 10001):
    print("%d %d" % (n, next(i)))
# David Fifield (david(AT)bamsoftware.com), Jan 17 2009

Formula

Conjecture: a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2; a(n) = (n+1) - a(a(n-1)+1) for n>=4. [Conjecture has been proved; see Letouzey link. - Pierre Letouzey, Sep 09 2015]

Conjecture: a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2; for n>=4, let T = n-a(n-1). If T = (n-1) - a((n-1)-1) then a(n) = max{a-inv(T)} otherwise a(n) = min{a-inv(T)}, where a-inv(n) is the inverse of a(n).

From Pierre Letouzey, Sep 09 2015: (Start)

Alternative definition via differences:

a(0) = 0; a(n+1) = a(n) + d(n) where:

d(0) = 1, d(1) = 0, d(2) = 1, d(3) = 1; d(n)=1-d(n-1)*d(a(n)) for n>=4.

Also satisfies: a(n-1)+a(a(n)) = n for n>=4.

a(n) = A005206(n) + e(n), where e(n) is 0 or 1 depending of the shape of the decomposition of n as sums of Fibonacci numbers via Zeckendorf's theorem. (End)

There is an 11-state Fibonacci automaton that generates the sequence a(n) as follows: the inputs are the Zeckendorf representations of n and x in parallel, and the automaton accepts if and only if x = a(n). See the file a123070.pdf in the "links" section above. - Jeffrey Shallit, Feb 19 2025

a(n) = A005206(n - 3) + 2, for n >= 3; a(n) = n - floor((n - 2)/(1 + phi)) - 1, where phi is the golden ratio (1 + sqrt(5))/2 = (A001622). - Alan Michael Gómez Calderón, Aug 05 2025