A066182 Permutation of the integers with cycle form {1}, {3, 2}, {6, 5, 4}, {10, 9, 8, 7}, ...

1, 3, 2, 6, 4, 5, 10, 7, 8, 9, 15, 11, 12, 13, 14, 21, 16, 17, 18, 19, 20, 28, 22, 23, 24, 25, 26, 27, 36, 29, 30, 31, 32, 33, 34, 35, 45, 37, 38, 39, 40, 41, 42, 43, 44, 55, 46, 47, 48, 49, 50, 51, 52, 53, 54, 66, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 78, 67, 68, 69, 70, 71

Offset: 1

Author

Wouter Meeussen, Dec 15 2001

Keywords

• easy
• nonn
• tabl

Comments

Arrange natural numbers 1,2,3,4,5,... as a triangle like A000027, then rotate each row of triangle one step right. - Antti Karttunen, May 07 2002

As a rectangular array, a(n) is the natural interspersion of the sequence of triangular numbers; see A192872. [Clark Kimberling, Aug 12 2011]

Examples

			Northwest corner, when sequence is formatted as the natural interspersion of the sequence (1,3,6,10,15,...) of triangular numbers:
  1...3...6...10...15
  2...4...7...11...16
  5...8...12..17...23
  9...13..18..24...31     [ _Clark Kimberling_, Aug 12 2011 ]
		

Links

• Matthew House, Table of n, a(n) for n = 1..45150
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse permutation: A066181.

Cf. A000027, A192872.

Programs

Mathematica

FromCycles[Table[n(n-1)/2+Range[n, 1, -1], {n, 13}]]

Python

from math import isqrt, comb
def A066182(n): return -1+n+comb(isqrt(n<<3)+3>>1,2)-comb(isqrt(n-1<<3)+3>>1,2) # Chai Wah Wu, Jun 09 2025

Formula

a(n) = -1+n+binomial(A002024(n)+1,2)-binomial(A002024(n-1)+1,2) where A002024(n) is round(sqrt(2*n)). - Brian Tenneson, Feb 03 2017