A208234 Second inverse function (numbers of columns) for pairing function A188568.
1, 2, 1, 1, 2, 3, 4, 2, 3, 1, 1, 4, 3, 2, 5, 6, 2, 4, 3, 5, 1, 1, 6, 3, 4, 5, 2, 7, 8, 2, 6, 4, 5, 3, 7, 1, 1, 8, 3, 6, 5, 4, 7, 2, 9, 10, 2, 8, 4, 6, 5, 7, 3, 9, 1, 1, 10, 3, 8, 5, 6, 7, 4, 9, 2, 11
Offset: 1
Keywords
Examples
The start of the sequence as triangle array read by rows: 1; 2,1; 1,2,3; 4,2,3,1; 1,4,3,2,5; 6,2,4,3,5,1; 1,6,3,4,5,2,7; ... 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.