A331105 - OEIS

%I #30 Jul 14 2023 11:54:15

%S 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,

%T 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,

%U 27,1,6,10,13,15,16,0,63,61,58,54,49,43,36,28,19,9

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

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

%H Alois P. Heinz, <a href="/A331105/b331105.txt">Rows n = 0..15, flattened</a>

%e Triangle T(n,k) begins:

%e   0;

%e   0,  1;

%e   0,  3,  1,  2;

%e   0,  7,  5,  2, 6, 1,  3, 4;

%e   0, 15, 13, 10, 6, 1, 11, 4, 12, 3, 9, 14, 2, 5, 7, 8;

%e   ...

%p T:= n-> (p-> seq(modp(-k*(k+1)/2, p), k=0..p-1))(2^n):

%p seq(T(n), n=0..6);

%p # second Maple program:

%p T:= proc(n, k) option remember;

%p       `if`(k=0, 0, T(n, k-1)-k mod 2^n)

%p     end:

%p seq(seq(T(n, k), k=0..2^n-1), n=0..6);

%t T[n_, k_] := T[n, k] = If[k == 0, 0, Mod[T[n, k - 1] - k, 2^n]];

%t Table[Table[T[n, k], {k, 0, 2^n - 1}], {n, 0, 6}] // Flatten (* _Jean-François Alcover_, Mar 28 2022, after _Alois P. Heinz_ *)

%Y Columns k=0-2 give: A000004, A000225, A036563 (for n>1).

%Y Row sums give A006516.

%Y Row lengths give A000079.

%Y T(n,n) gives A014833 (for n>0).

%Y T(n,2^n-1) gives A131577.

%Y Cf. A329278, A363674.

%K nonn,look,tabf

%O 0,5

%A _Alois P. Heinz_, Jan 09 2020