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a(n) is the smallest nonnegative number not in a(0..n-1) such that the sequence a(0..n) forms the starting row of an XOR-triangle with only distinct values in each row.
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%I #111 Sep 02 2023 02:25:52

%S 0,1,2,4,3,7,5,8,16,6,10,17,13,24,18,32,22,9,40,21,28,11,35,45,64,20,

%T 31,68,23,36,65,14,128,33,26,56,61,75,19,129,77,102,92,46,121,147,190,

%U 58,119,67,200,78,139,38,25,96,256,49,76,138,34,66,265,207,184,268

%N a(n) is the smallest nonnegative number not in a(0..n-1) such that the sequence a(0..n) forms the starting row of an XOR-triangle with only distinct values in each row.

%C This sequence is conjectured to be a permutation of the nonnegative integers.

%C The second row of the XOR-triangle ((a(0) XOR a(1)), (a(1) XOR a(2)), ...) is a permutation of the positive integers. Further rows miss additional numbers. The first diagonal of this triangle, 0, 1, 2, 7, 3, 5, ..., (A345237) is not a permutation, because it contains multiple equal values.

%C If we enumerate the imaginary units of a Cayley-Dickson algebra of order m by the positive integers 1 .. 2^m-1, XOR of any pair of these numbers represents the multiplication table of this algebra if signs are ignored. This explains why a(2^n) and A345237(2^n) are equal.

%C The antidiagonals starting at a(0..2) are {0}, {1, 1}, {2, 3, 2}. Let D(n) be the antidiagonal starting at a(n). Then XOR-sum(D(2^n-1)) = 0 and XOR-sum(D((2*m + 1)*2^(n+1)-1)) = XOR-sum(D((2*m + 1)*2^(p+1)-1)). XOR-sum means XOR over all elements. This property holds also for the XOR-triangle based on the nonnegative integers ordered in sequence.

%H Rémy Sigrist, <a href="/A346298/b346298.txt">Table of n, a(n) for n = 0..7448</a>

%H Thomas Scheuerle, <a href="/A346298/a346298.pdf">Triangle based on n = 0..149, drawn as colored surfaces bitwise for bit 0-7</a>. This shows interesting Sierpinski-like structures.

%H Thomas Scheuerle, <a href="/A346298/a346298.svg">Triangle based on n = 0..149, colored by number</a>.

%H Thomas Scheuerle, <a href="/A346298/a346298_1.pdf">Triangle based on n = 0..149, colored by parity</a>. Interestingly the amounts of red and green are approximately equal.

%H Rémy Sigrist, <a href="/A346298/a346298.txt">C++ program</a>.

%F a(2^n) = A345237(2^n).

%F a(2^n + m) XOR a(m) = A345237(2^p + q) XOR A345237(q) if 2^n + m = 2^p + q.

%F a(2^m + 2^n + 2^p + ...) = A345237(k) XOR A345237(k - 2^m) XOR A345237(k - 2^n) XOR A345237(k - 2^p) XOR A345237(k - 2^m - 2^n) XOR A345237(k - 2^m - 2^p) XOR A345237(k - 2^m - 2^n - 2^p) XOR ..., k = 2^m + 2^n + 2^p + ... .

%F Sum_{k=0..n} a(k) <= Sum_{k=0..n} A345237(k).

%F ( Sum_{k=0..n} a(k) + Sum_{k=0..n} A345237(k) )^0.4202... < n and > n - 30 at least for n < 500.

%e 0 1 2 4 3 7 5 8 16 ... <-- sequence

%e \ / \ / \ / \ / \ / \ / \ / \ / a(0),a(1),a(2),...

%e 1 3 6 7 4 2 13 24 ... <----------+

%e \ / \ / \ / \ / \ / \ / \ / |

%e 2 5 1 3 6 15 21 ... <-+ 2nd row is

%e \ / \ / \ / \ / \ / \ / | a(0) XOR a(1),

%e 7 4 2 5 9 26 ... | a(1) XOR a(2),

%e \ / \ / \ / \ / \ / | a(2) XOR a(3),

%e 3 6 7 12 19 ... | etc.

%e \ / \ / \ / \ / |

%e 5 1 11 31 ... |

%e \ / \ / \ / |

%e 4 10 20 ... |

%e \ / \ / |

%e 14 30 ... 3rd row is

%e \ / (a(0) XOR a(1)) XOR (a(1) XOR a(2)),

%e 16 ... (a(1) XOR a(2)) XOR (a(2) XOR a(3)),

%e etc.

%e We show why a(2^n) = A345237(2^n) by reproducing the same process with a randomly chosen set of octonion units: {e0,e1,e2,e5,e6}. XOR is replaced by multiplication.

%e e0 e1 e2 e5 e6

%e \/ \/ \/ \/

%e e1 e3 e7 -e3

%e \/ \/ \/

%e -e2 -e4 -e4

%e \/ \/

%e -e6 -e0

%e \/

%e e6

%o (MATLAB)

%o function a = A346298(max_n)

%o a(1) = 0;

%o for n = 1:max_n

%o t = 1;

%o while ~isok([a t])

%o t = t+1;

%o end

%o a = [a t];

%o end

%o end

%o function [ ok ] = isok( in )

%o ok = (length(in) == length(unique(in)));

%o x = in;

%o if ok

%o for k = 1:(length(x)-1)

%o x = bitxor(x(1:end-1),x(2:end));

%o ok = ok && (length(x) == length(unique(x)));

%o if ~ok

%o break;

%o end

%o end

%o end

%o end

%o (C++) See Links section.

%Y Cf. A345237, A338047 (XOR binomial transform).

%K nonn,base

%O 0,3

%A _Thomas Scheuerle_, Jul 13 2021