A358919 a(0) = 0, and for any n >= 0, a(n+1) is the sum of the lengths of the runs of consecutive terms a(i), ..., a(j) with 0 <= i <= j <= n such that a(i) XOR ... XOR a(j) = a(n) (where XOR denotes the bitwise XOR operator).

0, 1, 3, 1, 4, 1, 5, 5, 10, 4, 12, 18, 1, 13, 8, 22, 44, 7, 52, 1, 19, 35, 10, 43, 53, 7, 68, 1, 31, 24, 56, 73, 8, 126, 105, 35, 71, 36, 71, 60, 70, 1, 124, 180, 10, 172, 41, 182, 40, 288, 1, 232, 15, 201, 4, 271, 6, 213, 1, 233, 14, 230, 25, 216, 9, 157, 115

Offset: 0

Author

Rémy Sigrist, Dec 06 2022

Keywords

• nonn
• base

Comments

The sequence is unbounded (if the sequence was bounded, with greatest value m, then, by the pigeonhole principle, some value, say v, would appear infinitely many times, and the next value after the (m+1)-th occurrence of v would be > m, a contradiction).

Examples

			The first terms, alongside the corresponding pairs (i,j), are:
  n   a(n)  (i,j)'s
  --  ----  ---------------------------------
   0     0  N/A
   1     1  (0,0)
   2     3  (0,1), (1,1)
   3     1  (2,2)
   4     4  (0,1), (1,1), (3,3)
   5     1  (4,4)
   6     5  (0,1), (1,1), (3,3), (5,5)
   7     5  (3,4), (4,5), (6,6)
   8    10  (3,4), (4,5), (4,7), (6,6), (7,7)
   9     4  (6,8), (8,8)
  10    12  (3,5), (3,7), (4,4), (5,6), (9,9)
  11    18  (0,8), (1,8), (10,10)
  12     1  (11,11)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..15722
• Rémy Sigrist, C program
• Rémy Sigrist, Scatterplot of the first 350000 terms

Crossrefs

Cf. A358799, A358918.

Programs

C

See Links section.