A360363 -id:A360363 - cp's OEIS frontend

A360364 Triangle T(n, k), n > 0, k = 1..n, read by rows; T(n, k) = A360363(n+1) XOR A360363(k) (where XOR denotes the bitwise XOR operator).

3, 2, 1, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 8, 4, 17, 18, 19, 20, 24, 28, 33, 34, 35, 36, 40, 44, 48, 49, 50, 51, 52, 56, 60, 32, 16, 65, 66, 67, 68, 72, 76, 80, 96, 112, 84, 87, 86, 81, 93, 89, 69, 117, 101, 21, 107, 104, 105, 110, 98, 102, 122, 74, 90, 42, 63

Offset: 1

Rémy Sigrist, Feb 04 2023

All terms are distinct.
Every positive integer appears in this sequence:
- each time a power of 2 appears in A360363, say A360363(n) = 2^k,
- if the least value v missing from the bitwise XOR of two distinct terms among the first n terms of A360363 satisfies v < 2^k,
- then A360363(n+1) = 2^k + v and T(n,n) = v.

			Table begins:
    3,
    2, 1,
    5, 6, 7,
    9, 10, 11, 12,
    13, 14, 15, 8, 4,
    17, 18, 19, 20, 24, 28,
    33, 34, 35, 36, 40, 44, 48,
    49, 50, 51, 52, 56, 60, 32, 16,
    65, 66, 67, 68, 72, 76, 80, 96, 112,
    84, 87, 86, 81, 93, 89, 69, 117, 101, 21,
    107, 104, 105, 110, 98, 102, 122, 74, 90, 42, 63,
    129, 130, 131, 132, 136, 140, 144, 160, 176, 192, 213, 234,
    151, 148, 149, 146, 158, 154, 134, 182, 166, 214, 195, 252, 22,
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..7021
Rémy Sigrist, Scatterplot of (n, T(n, k)) for n = 1..1227, k = 1..n
Rémy Sigrist, Colored representation of the first 1227 rows (where the hue is function of T(n, k))
Rémy Sigrist, C++ program
Index entries for sequences that are permutations of the natural numbers

Cf. A360363.

A360706 a(n) is the least positive number not yet used such that its binary representation has either all or none of its 1-bits in common with the XOR of a(1) to a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 8, 12, 5, 10, 6, 9, 7, 16, 17, 24, 14, 11, 18, 20, 13, 15, 19, 32, 36, 21, 25, 26, 22, 23, 27, 33, 37, 28, 34, 30, 29, 40, 42, 31, 64, 96, 35, 68, 38, 44, 41, 43, 39, 48, 56, 45, 65, 66, 46, 47, 67, 80, 52, 49, 57, 50, 82, 69, 97, 51, 53, 60, 54, 55, 58, 72, 73, 59, 61, 76, 70, 71, 74

Offset: 1

Thomas Scheuerle, Feb 17 2023

The lexicographically earliest permutation of positive numbers such that the nim-sum of the first k elements equals the nim-sum of k-1 elements with the element at position k either arithmetically added or subtracted.
The first occurrence of a number m >= 2^k is always m = 2^k.
All positive integers will appear in this sequence: see link section for details.

			   n    a(n)  a(n) in binary   a(1) XOR ... XOR a(n-1) in binary
------------------------------------------------------------------
   1     1          1b             0b
   2     2         10b             1b
   3     3         11b            11b
   4     4        100b             0b
   5     8       1000b           100b
   6    12       1100b          1100b
   7     5        101b             0b
...
Signed version of this sequence such that the arithmetic sum over the first k values equals the nim-sum over the first k values of the original sequence:
1, 2, -3, 4, 8, -12, 5, 10, -6, -9, 7, 16, -17, 24, -14, 11, -18, 20, -13, ...

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, Scatterplot red: a(n) blue: a(1) XOR...XOR a(n-1) for n = 1...50000
Thomas Scheuerle, Proof that all positive integers will appear in this sequence
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to binary expansion of n

Cf. A000120, A116624, A269868, A330647, A360363, A375856.

function a = A360706( max_n )
    s = 0; a = []; t = [1:max_n];
    for n = 1:max_n
        k = 1;
        while (t(k) ~= bitand(s,t(k)))&&(0 ~= bitand(s,t(k)))
            k = k+1;
        end
        s = bitxor(s,t(k));
        a(n) = t(k);
        t(k) = max(t)+1; t = sort(t);
    end
end

{ m = s = 0; for (n = 1, 77, for (v = 1, oo, if (!bittest(s, v), x = bitand(m, v); if (x==0 || x==v, s += 2^v; m = bitxor(m, v); print1 (v", "); break;);););); } \\ Rémy Sigrist, Aug 31 2024

If a(m1) = 2^k and a(m2) = 2^k-1 then m1 - 2^k < 0 and m2 - (2^k-1) > 0 for k > 2.

cp's OEIS Frontend

A360364 Triangle T(n, k), n > 0, k = 1..n, read by rows; T(n, k) = A360363(n+1) XOR A360363(k) (where XOR denotes the bitwise XOR operator).

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A360706 a(n) is the least positive number not yet used such that its binary representation has either all or none of its 1-bits in common with the XOR of a(1) to a(n-1).

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