OFFSET
0,7
COMMENTS
If (x, y) and (x', y') are adjacent points on the trajectory of the map then max(|x - x'|, |y - y'|) is always 1 whereas for the Cantor enumeration this quantity can become arbitrarily large. In this sense our boustrophedonic variant is continuous whereas Cantor's realization is not.
We implemented the recursive enumeration as a state machine with two states to avoid the evaluation of the square root function.
The inverse function, computing n for given (x, y), is (x + y)*(x + y + 1)/2 + p where p = x if x - y is odd and y otherwise.
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, 2018.
A. L. Rosenberg, Allocating storage for extendible Arrays, J. ACM, vol 21(4), 1974, p. 652-670.
EXAMPLE
The map starts, for n = 0, 1, 2, ...
(0, 0), (0, 1), (1, 0), (2, 0), (1, 1), (0, 2), (0, 3), (1, 2), (2, 1), (3, 0),
(4, 0), (3, 1), (2, 2), (1, 3), (0, 4), (0, 5), (1, 4), (2, 3), (3, 2), (4, 1),
(5, 0), (6, 0), (5, 1), (4, 2), (3, 3), (2, 4), (1, 5), (0, 6), (0, 7), (1, 6),
(2, 5), (3, 4), (4, 3), (5, 2), (6, 1), (7, 0), ...
The enumeration can be seen as diagonal stripes layering on the origin:
(0, 0),
(0, 1), (1, 0),
(2, 0), (1, 1), (0, 2),
(0, 3), (1, 2), (2, 1), (3, 0),
(4, 0), (3, 1), (2, 2), (1, 3), (0, 4),
(0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0),
(6, 0), (5, 1), (4, 2), (3, 3), (2, 4), (1, 5), (0, 6),
(0, 7), (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1), (7, 0)
PROG
(Julia)
function A319571(n)
k, r = divrem(n, 2)
d = div(isqrt(8k+1) - 1, 2)
s = k - div(d*(d + 1), 2)
isodd(d) ? (s, d-s)[r+1] : (d-s, s)[r+1]
end
function stripe(x, y, state)
x == 0 && !state && return x, y+1, !state
y == 0 && state && return x+1, y, !state
state && return x+1, y-1, state
return x-1, y+1, state
end
function StripeEnumeration(len)
x, y, state = 0, 0, false
for n in 0:len
println("$n -> ($x, $y)")
x, y, state = stripe(x, y, state)
end
end
function Pairing(x, y)
p = isodd(x - y) ? x : y
div((x + y)*(x + y + 1), 2) + p
end
StripeEnumeration(40)
(Python)
from itertools import count, islice
def A319571_gen(): # generator of terms
for n in count(0):
for m in range(n+1):
yield from (m, n-m) if n % 2 else (n-m, m)
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Sep 23 2018
STATUS
approved