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

1,2

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; n - (t - 1)^2} + mod(t + 1; 2)*min{t; t^2 - n + 1}, 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;
  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).

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 *)

Prog

(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)

Crossrefs

Cf. A081344.

Sequence in context: A242357 A120423 A113137 * A238404 A331910 A240168

Adjacent sequences: A220600 A220601 A220602 * A220604 A220605 A220606

Keyword

nonn,tabf

Author

Boris Putievskiy, Dec 16 2012

Status

approved