A319514 The shell enumeration of N X N where N = {0, 1, 2, ...}, also called boustrophedonic Rosenberg-Strong function. Terms are interleaved x and y coordinates.

0, 0, 0, 1, 1, 1, 1, 0, 2, 0, 2, 1, 2, 2, 1, 2, 0, 2, 0, 3, 1, 3, 2, 3, 3, 3, 3, 2, 3, 1, 3, 0, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, 3, 4, 2, 4, 1, 4, 0, 4, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 5, 4, 5, 3, 5, 2, 5, 1, 5, 0, 6, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6, 5

Offset: 0

Author

Peter Luschny, Sep 22 2018

Keywords

• nonn

Comments

If (x, y) and (x', y') are adjacent points on the trajectory of the map then for the boustrophedonic Rosenberg-Strong function max(|x - x'|, |y - y'|) is always 1 whereas for the Rosenberg-Strong function this quantity can become arbitrarily large. In this sense the boustrophedonic variant is continuous in contrast to the original Rosenberg-Strong function.

We implemented the enumeration also as a state machine to avoid the evaluation of the square root function.

The inverse function, computing n for given (x, y), is m*(m + 1) + (-1)^(m mod 2)*(y - x) where m = max(x, y).

Examples

			The map starts, for n = 0, 1, 2, ...
(0, 0), (0, 1), (1, 1), (1, 0), (2, 0), (2, 1), (2, 2), (1, 2), (0, 2), (0, 3),
(1, 3), (2, 3), (3, 3), (3, 2), (3, 1), (3, 0), (4, 0), (4, 1), (4, 2), (4, 3),
(4, 4), (3, 4), (2, 4), (1, 4), (0, 4), (0, 5), (1, 5), (2, 5), (3, 5), (4, 5),
(5, 5), (5, 4), (5, 3), (5, 2), (5, 1), (5, 0), (6, 0), (6, 1), (6, 2), (6, 3),
(6, 4), (6, 5), (6, 6), (5, 6), (4, 6), (3, 6), (2, 6), (1, 6), (0, 6), ...
The enumeration can be seen as shells growing around the origin:
(0, 0);
(0, 1), (1, 1), (1, 0);
(2, 0), (2, 1), (2, 2), (1, 2), (0, 2);
(0, 3), (1, 3), (2, 3), (3, 3), (3, 2), (3, 1), (3, 0);
(4, 0), (4, 1), (4, 2), (4, 3), (4, 4), (3, 4), (2, 4), (1, 4), (0, 4);
(0, 5), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (5, 4), (5, 3), (5, 2),(5,1),(5,0);
		

References

• A. L. Rosenberg, H. R. Strong, Addressing arrays by shells, IBM Technical Disclosure Bulletin, vol 14(10), 1972, p. 3026-3028.

Links

• Peter Luschny, Table of n, a(n) for n = 0..10000
• Georg Cantor, Ein Beitrag zur Mannigfaltigkeitslehre, Journal für die reine und angewandte Mathematik 84 (1878), 242-258.
• Steven Pigeon, Mœud deux, 2018.
• A. L. Rosenberg, Allocating storage for extendible Arrays, J. ACM, vol 21(4), 1974, p. 652-670.
• M. P. Szudzik, The Rosenberg-Strong Pairing Function, arXiv:1706.04129 [cs.DM], 2017.

Crossrefs

Cf. A319289 (x coordinates), A319290 (y coordinates).

Cf. A319571 (stripe enumeration), A319572 (stripe x), A319573 (stripe y).

A319513 uses the encoding 2^x*3*y.

Cf. A038566/A038567, A157807/A157813.

Programs

Julia

function A319514(n)
    k, r = divrem(n, 2)
    m = x = isqrt(k)
    y = k - x^2
    x <= y && ((x, y) = (2x - y, x))
    isodd(m) ? (y, x)[r+1] : (x, y)[r+1]
end
[A319514(n) for n in 0:52] |> println
# The enumeration of N X N with a state machine:
# PigeonRosenbergStrong(n)
function PRS(x, y, state)
    x == 0 && state == 0 && return x, y+1, 1
    y == 0 && state == 2 && return x+1, y, 3
    x == y && state == 1 && return x, y-1, 2
    x == y && return x-1, y, 0
    state == 0 && return x-1, y, 0
    state == 1 && return x+1, y, 1
    state == 2 && return x, y-1, 2
    return x, y+1, 3
end
function ShellEnumeration(len)
    x, y, state = 0, 0, 0
    for n in 0:len
        println("$n -> ($x, $y)")
        x, y, state = PRS(x, y, state)
    end
end
# Computes n for given (x, y).
function Pairing(x::Int, y::Int)
    m = max(x, y)
    d = isodd(m) ? x - y : y - x
    m*(m + 1) + d
end
ShellEnumeration(20)