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

0,3

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).

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, C program

Rémy Sigrist, Scatterplot of the first 250000 terms

Example

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)

Prog

(C) See Links section.

Crossrefs

Cf. A331614, A332518.

Sequence in context: A358103 A112382 A117384 * A125160 A359027 A366691

Adjacent sequences: A358796 A358797 A358798 * A358800 A358801 A358802

Keyword

nonn,base

Author

Rémy Sigrist, Dec 06 2022

Status

approved