A329278 -id:A329278 - cp's OEIS frontend

A330766 Length of longest cycle in n-th row of A329278 (triangular numbers mod 2^n).

1, 1, 2, 6, 14, 30, 40, 55, 247, 488, 818, 1652, 4060, 3754, 15748, 20161, 20128, 85861, 159827, 211265, 620076, 993084, 1487646, 2542051, 12137774, 28169497, 32223531, 87591110, 232647749, 379598603, 877039442

Offset: 0

Peter Kagey, Dec 30 2019

nonn

			For n = 3, the third row of A329278 is [0, 1, 3, 6, 2, 7, 5, 4] (zero-indexed) which has cycle 2 -> 3 -> 6 -> 5 -> 7 -> 4 -> 2, a cycle of length a(3) = 6.

Mathematics Stack Exchange user joriki, Why do the triangular numbers initially form long cycles mod 2^k?

Cf. A329278.

A331105 T(n,k) = -k*(k+1)/2 mod 2^n; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 2, 0, 7, 5, 2, 6, 1, 3, 4, 0, 15, 13, 10, 6, 1, 11, 4, 12, 3, 9, 14, 2, 5, 7, 8, 0, 31, 29, 26, 22, 17, 11, 4, 28, 19, 9, 30, 18, 5, 23, 8, 24, 7, 21, 2, 14, 25, 3, 12, 20, 27, 1, 6, 10, 13, 15, 16, 0, 63, 61, 58, 54, 49, 43, 36, 28, 19, 9

Offset: 0

Alois P. Heinz, Jan 09 2020

Row n is a permutation of {0, 1, ..., A000225(n)}.

			Triangle T(n,k) begins:
  0;
  0,  1;
  0,  3,  1,  2;
  0,  7,  5,  2, 6, 1,  3, 4;
  0, 15, 13, 10, 6, 1, 11, 4, 12, 3, 9, 14, 2, 5, 7, 8;
  ...

Alois P. Heinz, Rows n = 0..15, flattened

Columns k=0-2 give: A000004, A000225, A036563 (for n>1).
Row sums give A006516.
Row lengths give A000079.
T(n,n) gives A014833 (for n>0).
T(n,2^n-1) gives A131577.
Cf. A329278, A363674.

T:= n-> (p-> seq(modp(-k*(k+1)/2, p), k=0..p-1))(2^n):
seq(T(n), n=0..6);
# second Maple program:
T:= proc(n, k) option remember;
      `if`(k=0, 0, T(n, k-1)-k mod 2^n)
    end:
seq(seq(T(n, k), k=0..2^n-1), n=0..6);

T[n_, k_] := T[n, k] = If[k == 0, 0, Mod[T[n, k - 1] - k, 2^n]];
Table[Table[T[n, k], {k, 0, 2^n - 1}], {n, 0, 6}] // Flatten (* Jean-François Alcover, Mar 28 2022, after Alois P. Heinz *)

A363674 T(n,k) is the decimal equivalent of the n-bit inverted Gray code for k; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 1, 7, 6, 4, 5, 1, 0, 2, 3, 15, 14, 12, 13, 9, 8, 10, 11, 3, 2, 0, 1, 5, 4, 6, 7, 31, 30, 28, 29, 25, 24, 26, 27, 19, 18, 16, 17, 21, 20, 22, 23, 7, 6, 4, 5, 1, 0, 2, 3, 11, 10, 8, 9, 13, 12, 14, 15, 63, 62, 60, 61, 57, 56, 58, 59, 51, 50, 48

Offset: 0

Alois P. Heinz, Jun 14 2023

Row n is a permutation of {0, 1, ..., A000225(n)}.

			Triangle T(n,k) begins:
   0;
   1,  0;
   3,  2,  0,  1;
   7,  6,  4,  5, 1, 0,  2,  3;
  15, 14, 12, 13, 9, 8, 10, 11, 3, 2, 0, 1, 5, 4, 6, 7;
  ...
T(n,k) written in n-bit binary begins:
    ();
     1,    0;
    11,   10,   00,   01;
   111,  110,  100,  101,  001,  000,  010,  011;
  1111, 1110, 1100, 1101, 1001, 1000, 1010, 1011, 0011, 0010, 0000, ...;
  ...

Alois P. Heinz, Rows n = 0..14, flattened
Wikipedia, Gray code

Columns k=0-2 give: A000225, A000918 (for n>=1), A028399 (for n>=2).
Row sums give A006516.
Cf. A000120, A000975, A003188, A063524, A097072, A236840, A329278, A331105, A362160.

T:= (n, k)-> Bits[Xor](2^n-1-k, iquo(k, 2)):
seq(seq(T(n, k), k=0..2^n-1), n=0..6);

T(n,k) = 2^n - 1 - A003188(k) = A000225(n) - A003188(k).
Sum_{k=0..2^n-1} (-1)^k * T(n,k) = A063524(n).
T(n,0) = T(n+1,2^(n+1)-1) = A000225(n).
T(n,A000975(n)) = 0.
T(n,A097072(n)) = 1 for n >= 1.
T(n,k) = T(n-1,k) + 2^(n-1) for n >= 1 and 0 <= k < 2^(n-1).
T(n,k) = T(n-1,2^n-1-k) for n >= 1 and 2^(n-1) <= k < 2^n.
A000120(T(n,n)) = A236840(n).

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A330766 Length of longest cycle in n-th row of A329278 (triangular numbers mod 2^n).

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A331105 T(n,k) = -k*(k+1)/2 mod 2^n; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

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A363674 T(n,k) is the decimal equivalent of the n-bit inverted Gray code for k; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

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