A371214 Lexicographically earliest sequence of distinct nonnegative integers such that for any n > 0, a(n-1) + a(n) is a multiple of n, and the least value not yet in the sequence appears as soon as possible.

0, 1, 7, 2, 22, 3, 45, 4, 76, 5, 115, 6, 18, 8, 216, 9, 279, 10, 350, 11, 429, 12, 516, 13, 611, 14, 714, 15, 825, 16, 944, 17, 47, 19, 1205, 20, 1348, 21, 55, 23, 1657, 24, 1824, 25, 1999, 26, 2182, 27, 2373, 28, 2572, 29, 2779, 30, 2994, 31, 3217, 32, 3448

Offset: 0

Author

Rémy Sigrist, Mar 15 2024

Keywords

• nonn

Comments

To build the sequence:

- we start with a(0) = 0, and repeatedly:

- let a(n) be the last known term and v the least value not yet in the sequence,

- if a(n) + v is a multiple of n+1 then a(n+1) = v,

- otherwise a(n+2) = v and a(n+1) is chosen as small as possible in such a way as to satisfy the required congruences (this is always possible as n+1 and n+2 are coprime).

The construction is similar to that of A367288.

This sequence is a variant of A099506 and, by design, is guaranteed to be a permutation of the nonnegative integers (with inverse A371215).

Examples

			The first terms are:
  n   a(n)  (a(n-1) + a(n))/n
  --  ----  -----------------
   0     0  N/A
   1     1                  1
   2     7                  4
   3     2                  3
   4    22                  6
   5     3                  5
   6    45                  8
   7     4                  7
   8    76                 10
   9     5                  9
  10   115                 12
  11     6                 11
  12    18                  2
		

Links

• Rémy Sigrist, PARI program
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A099506, A367288, A371215 (inverse).

Programs

PARI

See Links section.