A220604 Second inverse function (numbers of columns) for pairing function A081344.

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

Offset: 1

Author

Boris Putievskiy, Dec 17 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;
  1,2,2;
  3,3,3,2,1;
  1,2,3,4,4,4,4;
  ...
If k is even the row is 1,2,...,k,k...k (k times repetition "k" at the end of row).
If k is odd 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

Python

t=int(math.sqrt(n-1))+1
j=(t % 2)*min(t,t**2-n+1) + ((t+1) % 2)*min(t,n-(t-1)**2)

Formula

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