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 A346298 a(n) is the smallest nonnegative number not yet 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. 2
 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, 31, 68, 23, 36, 65, 14, 128, 33, 26, 56, 61, 75, 19, 129, 77, 102, 92, 46, 121, 147, 190, 58, 119, 67, 200, 78, 139, 38, 25, 96, 256, 49, 76, 138, 34, 66, 265, 207, 184, 268 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS This sequence is conjectured to be a permutation of the nonnegative integers. 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. 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. 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. LINKS Thomas Scheuerle, Triangle based on n = 0..149, drawn as colored surfaces bitwise for bit 0-7. This shows interesting Sierpinski-like structures. Thomas Scheuerle, Triangle based on n = 0..149, colored by number. Thomas Scheuerle, Triangle based on n = 0..149, colored by parity. Interestingly the amounts of red and green are approximately equal. FORMULA a(2^n) = A345237(2^n). a(2^n + m) XOR a(m) = A345237(2^p + q) XOR A345237(q) if 2^n + m = 2^p + q. 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 + ... . Sum_{k=0..n} a(k) <= Sum_{k=0..n} A345237(k). ( Sum_{k=0..n} a(k) + Sum_{k=0..n} A345237(k) )^0.4202... < n and > n - 30 at least for n < 500. EXAMPLE 0   1   2   4   3   7   5   8  16 ... <-- sequence    \ / \ / \ / \ / \ / \ / \ / \ /         a(0),a(1),a(2),...     1   3   6   7   4   2  13  24 ... <----------+      \ / \ / \ / \ / \ / \ / \ /                 |       2   5   1   3   6  15  21 ... <-+         2nd row is        \ / \ / \ / \ / \ / \ /        |       a(0) XOR a(1),         7   4   2   5   9  26 ...     |       a(1) XOR a(2),          \ / \ / \ / \ / \ /          |       a(2) XOR a(3),           3   6   7  12  19 ...       |            etc.            \ / \ / \ / \ /            |             5   1  11  31 ...         |              \ / \ / \ /              |               4  10  20 ...           |                \ / \ /                |                14  30 ...            3rd row is                  \ /     (a(0) XOR a(1)) XOR (a(1) XOR a(2)),                  16 ...  (a(1) XOR a(2)) XOR (a(2) XOR a(3)),                                          etc. 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.   e0  e1  e2  e5  e6     \/  \/  \/  \/     e1  e3  e7 -e3       \/  \/  \/      -e2 -e4 -e4         \/  \/        -e6 -e0           \/           e6 PROG (MATLAB) function a = A346298(max_n)     a(1) = 0;     for n = 1:max_n         t = 1;         while ~isok([a t])             t = t+1;         end         a = [a t];     end end function [ ok ] = isok( in )     ok = (length(in) == length(unique(in)));     x = in;     if ok         for k = 1:(length(x)-1)             x = bitxor(x(1:end-1), x(2:end));             ok = ok && (length(x) == length(unique(x)));             if ~ok                 break;             end         end     end end CROSSREFS Cf. A345237, A338047 (XOR binomial transform). Sequence in context: A191666 A215673 A120619 * A084385 A073885 A256283 Adjacent sequences:  A346295 A346296 A346297 * A346299 A346300 A346301 KEYWORD nonn,base AUTHOR Thomas Scheuerle, Jul 13 2021 STATUS approved

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Last modified October 23 16:20 EDT 2021. Contains 348215 sequences. (Running on oeis4.)