OFFSET
1,2
COMMENTS
Permutation of the 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.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Table read by boustrophedonic ("ox-plowing") method. Let m be natural number. The order of the list:
T(1,1)=1;
T(2,1), T(2,2), T(1,2);
. . .
T(1,2*m+1), T(2,2*m+1), ... T(2*m,2*m+1), T(2*m+1,2*m+1), T(2*m+1,2*m), ... T(2*m+1,1);
T(2*m,1), T(2*m,2), ... T(2*m,2*m-1), T(2*m,2*m), T(2*m-1,2*m), ... T(1,2*m);
. . .
The first row is layer read clockwise, the second row is layer counterclockwise.
LINKS
Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
FORMULA
As table
T(n,k) = n*n/2+4*(floor((k-1)/2)+1)*n+ceiling((k-1)^2/2), n,k > 0.
As linear sequence
a(n)= (m1+m2-1)*(m1+m2-2)/2+m1, where
m1=floor((i+j)/2) + floor(i/2)*(-1)^(2*i+j-1), m2=int((i+j+1)/2)+int(i/2)*(-1)^(2*i+j-2),
where i=(t mod 2)*min(t; n-(t-1)^2) + (t+1 mod 2)*min(t; t^2-n+1), j=(t mod 2)*min(t; t^2-n+1) + (t+1 mod 2)*min(t; n-(t-1)^2), t=floor(sqrt(n-1))+1.
EXAMPLE
The start of the sequence as table:
1....2...5...8..13..18...
3....4...9..12..19..24...
6....7..14..17..26..31...
10..11..20..23..34..39...
15..16..27..30..43..48...
21..22..35..38..53..58...
. . .
The start of the sequence as triangle array read by rows:
1;
3,4,2;
5,9,14,7,6;
10,11,20,23,17,12,8;
13,19,26,34,43,30,27,16,15;
21,22,35,38,53,58,48,39,31,24,18;
. . .
Row number r contains 2*r-1 numbers.
PROG
(Python)
t=int((math.sqrt(n-1)))+1
i=(t % 2)*min(t, n-(t-1)**2) + ((t+1) % 2)*min(t, t**2-n+1)
j=(t % 2)*min(t, t**2-n+1) + ((t+1) % 2)*min(t, n-(t-1)**2)
m1=int((i+j)/2)+int(i/2)*(-1)**(2*i+j-1)
m2=int((i+j+1)/2)+int(i/2)*(-1)**(2*i+j-2)
result=(m1+m2-1)*(m1+m2-2)/2+m1
CROSSREFS
KEYWORD
nonn
AUTHOR
Boris Putievskiy, Mar 11 2013
STATUS
approved