A127366 Let m = floor(sqrt(n)); if n and m have the same parity, a(n) = n + m, otherwise a(n) = n - m.

0, 2, 1, 4, 6, 3, 8, 5, 10, 12, 7, 14, 9, 16, 11, 18, 20, 13, 22, 15, 24, 17, 26, 19, 28, 30, 21, 32, 23, 34, 25, 36, 27, 38, 29, 40, 42, 31, 44, 33, 46, 35, 48, 37, 50, 39, 52, 41, 54, 56, 43, 58, 45, 60, 47, 62, 49, 64, 51, 66, 53, 68, 55, 70, 72, 57, 74, 59, 76, 61, 78, 63, 80

Offset: 0

Author

Franklin T. Adams-Watters, Jan 11 2007

Keywords

• nice
• nonn

Comments

This is a permutation of the nonnegative integers; it can also be generated by the rule (with m = floor(sqrt(n))): if n - m is not yet in the sequence, a(n) = n - m, otherwise a(n) = n + m. All cycles in this permutation are finite. There is one relatively large cycle starting at n = 4k^2 - 2k + 1 for each k and k 2-cycles for n = (2k - 1)^2 + 2i and (2k - 1)^2 + 2k - 1 + 2i with 0 <= i < k.

a(A133280(n,k)) mod 2 = 0 and a(A195437(n,k)) mod 2 = 1, 0 <= k < n. [Reinhard Zumkeller, Oct 12 2011]

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A127367.

Cf. A000196.

Programs

Haskell

a127366 n | even n'   = n'
          | otherwise = 2*n - n'
          where n' = n + a000196 n
-- Reinhard Zumkeller, Oct 12 2011

Mathematica

a[n_] := If[m = Floor[Sqrt[n]]; OddQ[n] && OddQ[m] || EvenQ[n] && EvenQ[m], n+m, n-m]; Table[ a[n], {n, 0, 72}](* Jean-François Alcover, Nov 30 2011 *)