A060030 a(1) = 1, a(2) = 2; thereafter a "hole" is defined to be any positive number not in the sequence a(1)..a(n-1) and less than the largest term; if there exists at least one hole, then a(n) is the largest hole, otherwise a(n) = a(n-2) + a(n-1).

1, 2, 3, 5, 4, 9, 8, 7, 6, 13, 12, 11, 10, 21, 20, 19, 18, 17, 16, 15, 14, 29, 28, 27, 26, 25, 24, 23, 22, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83

Offset: 1

Author

William Nelles (wnelles(AT)flashmail.com), Mar 17 2001

Keywords

• easy
• nice
• nonn

Comments

A self-inverse permutation of the natural numbers: a(a(n)) = n and a(n) <> n for n > 3. [Reinhard Zumkeller, Apr 29 2012]

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

See A060482 for successive records, A027383 for the final hole-filling values, A016116 for the difference between top and bottom of downward subsequences, A052551 for number of terms in downward subsequences.

Cf. A060000, A000045.

Programs

Haskell

import Data.List (delete)
a060030 n = a060030_list !! (n-1)
a060030_list = 1 : 2 : f 1 2 [3..] where
   f u v ws = y : f v y (delete y ws) where
     y = if null xs then u + v else last xs
     xs = takeWhile (< v) ws
-- Reinhard Zumkeller, Apr 29 2012

Mathematica

a[1] = 1; a[2] = 2;
a[n_] := a[n] = Module[{A, H}, A = Array[a, n-1]; H = Complement[ Range[a[n-1]], A]; If[H != {}, H[[-1]], a[n-2] + a[n-1]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 23 2024 *)

Extensions

Offset corrected by Reinhard Zumkeller, Apr 29 2012

Name made more explicit by Jean-François Alcover, Apr 23 2024