A209279 First inverse function (numbers of rows) for pairing function A185180.
1, 1, 2, 2, 1, 3, 2, 3, 1, 4, 3, 2, 4, 1, 5, 3, 4, 2, 5, 1, 6, 4, 3, 5, 2, 6, 1, 7, 4, 5, 3, 6, 2, 7, 1, 8, 5, 4, 6, 3, 7, 2, 8, 1, 9, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 6, 5, 7, 4, 8, 3, 9, 2, 10, 1, 11, 6, 7, 5, 8, 4, 9, 3, 10, 2, 11, 1, 12, 7, 6, 8, 5, 9, 4, 10, 3, 11, 2, 12, 1, 13
Offset: 1
Examples
The start of the sequence as table T(r,s) r,s >0 read by antidiagonals: 1...1...2...2...3...3...4...4... 2...1...3...2...4...3...5...4... 3...1...4...2...5...3...6...4... 4...1...5...2...6...3...7...4... 5...1...6...2...7...3...8...4... 6...1...7...2...8...3...9...4... 7...1...8...2...9...3..10...4... ... The start of the sequence as triangle array read by rows: 1; 1, 2; 2, 1, 3; 2, 3, 1, 4; 3, 2, 4, 1, 5; 3, 4, 2, 5, 1, 6; 4, 3, 5, 2, 6, 1, 7; 4, 5, 3, 6, 2, 7, 1, 8; ... Row number r contains permutation numbers form 1 to r. If r is odd (r+1)/2, (r+1)/2-1, (r+1)/2+1,...r-1, 1, r. If r is even r/2, r/2+1, r/2-1, ... r-1, 1, r.
Links
- Boris Putievskiy, Rows n = 1..140 of triangle, flattened
- Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
- Eric Weisstein's World of Mathematics, Pairing functions
Programs
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Mathematica
T[n_, k_] := Abs[(2*k - 1 + (-1)^(n - k)*(2*n + 1))/4]; Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *) -
PARI
T(n, k)=abs((2*k-1+(-1)^(n-k)*(2*n+1))/4) \\ Andrew Howroyd, Dec 31 2017
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Python
# Edited by M. F. Hasler, May 30 2020 def a(n): t = int((math.sqrt(8*n-7) - 1)/2); i = n-t*(t+1)/2; return int(t/2)+1+int(i/2)*(-1)**(i+t+1)
Formula
a(n) = |A128180(n)|.
a(n) = floor((t+2)/2) + floor(i/2)*(-1)^(i+t+1), where t=floor((-1+sqrt(8*n-7))/2), i=n-t*(t+1)/2.
T(r,2s)=s, T(r,2s-1)= r+s-1.(When read as table T(r,s) by antidiagonals.)
T(n,k) = ceiling((n + (-1)^(n-k)*k)/2) = (n+k)/2 if n-k even, otherwise (n-k+1)/2. - M. F. Hasler, May 30 2020
Extensions
Data corrected by Andrew Howroyd, Dec 31 2017
Comments