A014833 -id:A014833 - cp's OEIS frontend

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.

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 *)

A130104 Expansion of x(1-3x+5x^2-2x^3)/((1-x)^3*(1-2x)).

Original entry on oeis.org

0, 1, 2, 6, 17, 43, 100, 220, 467, 969, 1982, 4018, 8101, 16279, 32648, 65400, 130919, 261973, 524098, 1048366, 2096921, 4194051, 8388332, 16776916, 33554107, 67108513, 134217350, 268435050, 536870477, 1073741359

Offset: 0

Paul Barry, May 07 2007

a(n+1)=A014833(n+2).

Partial sums of A130103.

Seiichi Manyama, Table of n, a(n) for n = 0..3320
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A014833, A130103.

Accumulate[Join[{0,1},LinearRecurrence[{4,-5,2},{1,4,11},40]]] (* Harvey P. Dale, May 16 2014 *)
Join[{0},LinearRecurrence[{5, -9, 7, -2},{1, 2, 6, 17},29]] (* Ray Chandler, Aug 03 2015 *)

a(n)=2^(n+1)-C(n+2,2)-0^n.

cp's OEIS Frontend

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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