OFFSET
1,2
COMMENTS
A simple permutation of natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Jan 09 2013
LINKS
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
FORMULA
T(n+1, k)=n*n+k, T(k, n+1)=(n+1)*(n+1)+1-k, 1 <= k <= n+1.
a(n)=i^2-j+1 if i >= j, a(n)=(j-1)^2 + i if i < j, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jan 09 2013
EXAMPLE
1 4 9 16 .. => a(1)= 1
2 3 8 15 .. => a(2)= 2, a(3)=4
5 6 7 14 .. => a(4)= 5, a(5)=3, a(6)=9
10 11 12 13 .. => a(7)=10, a(8)=6, a(9)=8, a(10)=16
MATHEMATICA
Table[ If[n < 2*k-1, k^2 + k - n, (n-k)^2 + k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2013 *)
PROG
(Python)
t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i>=j:
result=i**2-j+1
else:
result=(j-1)**2+i
# Boris Putievskiy, Jan 09 2013
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Frank Ellermann, Apr 23 2001
STATUS
approved