OFFSET
1,2
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
a(n)=(m1+m2-1)*(m1+m2-2)/2+m1, where
m1=(3*i+j-1-(-1)^i+(i+j-2)*(-1)*t)/4,
m2=((1+(-1)^i)*((1+(-1)^j)*2*int((j+2)/4)-(-1+(-1)^j)*(2*int((i+4)/4)+2*int(j/2)))-(-1+(-1)^i)*((1+(-1)^j)*(1+2*int(i/4)+2*int(j/2))-(-1+(-1)^j)*(1+2*int(j/4))))/4,
i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
EXAMPLE
The start of the sequence as triangle array read by rows:
1;
4,2;
3,5,6;
11,7,8,12;
13,9,10,14,15;
22,16,17,23,24,18;
19,25,26,20,21,27,28;
. . .
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of above triangle array.
Last 2*r-1 numbers are from the row number 2*r-1 of above triangle array.
1;
4,2,3,5,6;
11,7,8,12,13,9,10,14,15;
22,16,17,23,24,18,19,25,26,20,21,27,28;
Row number r contains permutation of 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r: 2*r*r-3*r+2, 2*r*r-5*r+4, 2*r*r-5*r+5,... 2*r*r-r-1, 2*r*r-r.
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
m1=(3*i+j-1-(-1)**i+(i+j-2)*(-1)**(i+j))/4
m2=((1+(-1)**i)*((1+(-1)**j)*2*int((j+2)/4)-(-1+(-1)**j)*(2*int((i+4)/4)+2*int(j/2)))-(-1+(-1)**i)*((1+(-1)**j)*(1+2*int(i/4)+2*int(j/2))-(-1+(-1)**j)*(1+2*int(j/4))))/4
result=(m1+m2-1)*(m1+m2-2)/2+m1
CROSSREFS
KEYWORD
nonn
AUTHOR
Boris Putievskiy, Mar 01 2013
STATUS
approved