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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).

%I A358799 #12 Dec 12 2022 12:15:09
%S A358799 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,
%T A358799 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,
%U A358799 47,10,49,17,37,21,38,23,36,24,49,30,48,24,52,22,45
%N 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).
%C A358799 This sequence is a variant of A331614 and A332518; here we use binary XOR, there addition and multiplication, respectively.
%C A358799 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).
%H A358799 Rémy Sigrist, <a href="/A358799/b358799.txt">Table of n, a(n) for n = 0..10000</a>
%H A358799 Rémy Sigrist, <a href="/A358799/a358799.txt">C program</a>
%H A358799 Rémy Sigrist, <a href="/A358799/a358799.png">Scatterplot of the first 250000 terms</a>
%e A358799 The first terms, alongside the corresponding pairs (i,j)'s, are:
%e A358799   n   a(n)  (i,j)'s
%e A358799   --  ----  ---------------------------------------------------------
%e A358799    0     0  N/A
%e A358799    1     1  (0,0)
%e A358799    2     2  (0,1), (1,1)
%e A358799    3     1  (2,2)
%e A358799    4     3  (0,1), (1,1), (3,3)
%e A358799    5     4  (0,2), (1,2), (2,3), (4,4)
%e A358799    6     2  (2,5), (5,5)
%e A358799    7     5  (0,3), (1,3), (2,2), (3,4), (6,6)
%e A358799    8     4  (0,5), (1,5), (4,6), (7,7)
%e A358799    9     5  (2,5), (3,6), (4,8), (5,5), (8,8)
%e A358799   10     6  (0,5), (1,5), (3,8), (4,6), (7,7), (9,9)
%e A358799   11     8  (0,8), (1,8), (2,6), (3,5), (3,10), (5,6), (6,9), (10,10)
%e A358799   12     2  (6,11), (11,11)
%o A358799 (C) See Links section.
%Y A358799 Cf. A331614, A332518.
%K A358799 nonn,base
%O A358799 0,3
%A A358799 _Rémy Sigrist_, Dec 06 2022