A220603 First inverse function (numbers of rows) for pairing function A081344.

1, 2, 2, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Author

Boris Putievskiy, Dec 16 2012

Keywords

• nonn
• tabf

Examples

			The start of the sequence as triangle array T(n,k) read by rows, row number k contains 2k-1 numbers:
  1;
  2,2,1;
  1,2,3,3,3;
  4,4,4,4,3,2,1;
  ...
If k is odd the row is 1,2,...,k,k...k (k times repetition "k" at the end of row).
If k is even the row is k,k,...k,k-1,k-2,...1 (k times repetition "k" at the start of row).
		

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. A081344.

Programs

Mathematica

row[n_] := If[OddQ[n], Range[n-1]~Join~Table[n, {n}], Table[n, {n}]~Join~ Range[n-1, 1, -1]];
row /@ Range[10] // Flatten (* Jean-François Alcover, Nov 19 2019 *)

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)

Formula

As a linear array, the sequence is a(n) = mod(t;2)*min{t; n - (t - 1)^2} + mod(t + 1; 2)*min{t; t^2 - n + 1}, where t=floor[sqrt(n-1)]+1.