A208233 First inverse function (numbers of rows) for pairing function A188568.
1, 1, 2, 3, 2, 1, 1, 3, 2, 4, 5, 2, 3, 4, 1, 1, 5, 3, 4, 2, 6, 7, 2, 5, 4, 3, 6, 1, 1, 7, 3, 5, 4, 6, 2, 8, 9, 2, 7, 4, 5, 6, 3, 8, 1, 1, 9, 3, 7, 5, 6, 4, 8, 2, 10, 11, 2, 9, 4, 7, 6, 5, 8, 3, 10, 1
Offset: 1
Examples
The start of the sequence as triangle array read by rows: 1; 1,2; 3,2,1; 1,3,2,4; 5,2,3,41; 1,5,3,4,2,6; 7,2,5,4,3,6,1; ... Row number k contains permutation numbers form 1 to k.
Links
- Boris Putievskiy, Rows n = 1..140 of triangle, flattened
- Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Crossrefs
Cf. A188568.
Programs
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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= max(i,j)*((-1)**i+1)/2-min(i,j)*((-1)**i-1)/2 else: result=-max(i,j)*((-1)**j-1)/2+min(i,j)*((-1)**j+1)/2
Formula
a(n) = max(i,j)*((-1)^i+1)/2-min(i,j)*((-1)^i-1)/2, if i>=j
a(n) = -max(i,j)*((-1)^j-1)/2+min(i,j)*((-1)^j+1)/2, if i
where
t = floor((-1+sqrt(8*n-7))/2),
i = n-t*(t+1)/2,
j = (t*t+3*t+4)/2-n.