A358799 a(0) = 0, and for any n >= 0, a(n+1) is the number of ways to write a(n) = a(i) XOR ... XOR a(j) with 0 <= i <= j <= n (where XOR denotes the bitwise XOR operator).

0, 1, 2, 1, 3, 4, 2, 5, 4, 5, 6, 8, 2, 11, 2, 13, 6, 14, 10, 9, 9, 12, 14, 16, 2, 24, 6, 29, 5, 23, 3, 27, 12, 23, 9, 26, 17, 13, 26, 19, 15, 32, 4, 46, 2, 51, 1, 45, 6, 48, 6, 49, 7, 41, 9, 47, 10, 49, 17, 37, 21, 38, 23, 36, 24, 49, 30, 48, 24, 52, 22, 45

Offset: 0

Author

Rémy Sigrist, Dec 06 2022

Keywords

• nonn
• base

Comments

This sequence is a variant of A331614 and A332518; here we use binary XOR, there addition and multiplication, respectively.

This 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)'s, are:
  n   a(n)  (i,j)'s
  --  ----  ---------------------------------------------------------
   0     0  N/A
   1     1  (0,0)
   2     2  (0,1), (1,1)
   3     1  (2,2)
   4     3  (0,1), (1,1), (3,3)
   5     4  (0,2), (1,2), (2,3), (4,4)
   6     2  (2,5), (5,5)
   7     5  (0,3), (1,3), (2,2), (3,4), (6,6)
   8     4  (0,5), (1,5), (4,6), (7,7)
   9     5  (2,5), (3,6), (4,8), (5,5), (8,8)
  10     6  (0,5), (1,5), (3,8), (4,6), (7,7), (9,9)
  11     8  (0,8), (1,8), (2,6), (3,5), (3,10), (5,6), (6,9), (10,10)
  12     2  (6,11), (11,11)
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10000
• Rémy Sigrist, C program
• Rémy Sigrist, Scatterplot of the first 250000 terms

Crossrefs

Cf. A331614, A332518.

Programs

C

See Links section.