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

1,3

Links

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

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.

Example

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).

Prog

(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)

Crossrefs

Cf. A081344.

Sequence in context: A283763 A357622 A357621 * A331484 A046918 A238732

Adjacent sequences: A220601 A220602 A220603 * A220605 A220606 A220607

Keyword

nonn,tabf

Author

Boris Putievskiy, Dec 17 2012

Status

approved