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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

Table of n, a(n) for n=0..65.

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.)