A106262 An invertible triangle of remainders of 2^n.

1, 0, 1, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 2, 1, 0, 1, 0, 4, 2, 1, 0, 2, 0, 3, 4, 2, 1, 0, 1, 0, 1, 2, 4, 2, 1, 0, 2, 0, 2, 4, 1, 4, 2, 1, 0, 1, 0, 4, 2, 2, 0, 4, 2, 1, 0, 2, 0, 3, 4, 4, 0, 8, 4, 2, 1, 0, 1, 0, 1, 2, 1, 0, 7, 8, 4, 2, 1, 0, 2, 0, 2, 4, 2, 0, 5, 6, 8, 4, 2, 1, 0, 1, 0, 4, 2, 4, 0, 1, 2, 5, 8, 4, 2, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = 2^(n-k) mod (k+2).

Sum_{k=0..n} T(n, k) = A106263(n) (row sums).

Sum_{k=0..floor(n/2)} T(n-k, k) = A106264(n) (diagonal sums).

From G. C. Greubel, Jan 10 2023: (Start)

T(n, 0) = A000007(n).

T(n, 1) = A000034(n+1).

T(2*n, n) = A213859(n).

T(2*n, n-1) = A015910(n+1).

T(2*n, n+1) = A294390(n+3).

T(2*n+1, n-1) = A112983(n+1).

T(2*n+1, n+1) = A294389(n+3).

T(2*n-1, n-1) = A062173(n+1). (End)

Example

Triangle begins:
  1;
  0, 1;
  0, 2, 1;
  0, 1, 2, 1;
  0, 2, 0, 2, 1;
  0, 1, 0, 4, 2, 1;
  0, 2, 0, 3, 4, 2, 1;
  0, 1, 0, 1, 2, 4, 2, 1;
  0, 2, 0, 2, 4, 1, 4, 2, 1;
  0, 1, 0, 4, 2, 2, 0, 4, 2, 1;

Mathematica

Table[PowerMod[2, n-k, k+2], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 10 2023 *)

Prog

(Magma) [Modexp(2, n-k, k+2): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 10 2023
(SageMath) flatten([[power_mod(2, n-k, k+2) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jan 10 2023

Crossrefs

Cf. A106263 (row sums), A106264 (diagonal sums).

Cf. A000034, A015910, A062173, A112983, A213859, A294389, A294390.

Sequence in context: A305259 A029370 A369572 * A025870 A104276 A216191

Adjacent sequences: A106259 A106260 A106261 * A106263 A106264 A106265

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 28 2005

Status

approved