A372344 Square array A(n, k), n, k >= 0, read by upwards antidiagonals; for any n, k >= 0 with respective binary expansions Sum_{i >= 0} b_i * 2^i and Sum_{i >= 0} c_i * 2^i, A(n, k) = Sum_{i >= 0} (b_{i+1} * c_i - b_i * c_{i+1}) * 3^i.

0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 3, 0, -3, 0, 0, 0, 1, 2, 3, -3, -2, -1, 0, 0, 1, 3, 2, 0, -2, -3, -1, 0, 0, 0, 2, 4, 0, 0, -4, -2, 0, 0, 0, 0, 0, 3, -3, 0, 3, -3, 0, 0, 0, 0, 1, -1, 0, -3, -2, 2, 3, 0, 1, -1, 0

Offset

0,24

Comments

The digits in the balanced ternary expansion of A(n, k) correspond to determinants of 2 X 2 matrices made up of binary digits of n and k.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10010

Rémy Sigrist, Scatterplot of (n, k) such that A(n, k) = 0 and n, k <= 2^10

Rémy Sigrist, Colored scatterplot of (n, k) with n, k <= 2^10 (where the color denotes the sign of A(n, k): blue, black and red for -1, 0 and +1, respectively)

Formula

A(k, n) = -A(n, k).

A(n, n) = 0.

Example

Array A(n, k) begins:
  n\k | 0  1   2   3   4   5   6   7   8   9  10
  ----+-----------------------------------------
    0 | 0  0   0   0   0   0   0   0   0   0   0
    1 | 0  0  -1  -1   0   0  -1  -1   0   0  -1
    2 | 0  1   0   1  -3  -2  -3  -2   0   1   0
    3 | 0  1  -1   0  -3  -2  -4  -3   0   1  -1
    4 | 0  0   3   3   0   0   3   3  -9  -9  -6
    5 | 0  0   2   2   0   0   2   2  -9  -9  -7
    6 | 0  1   3   4  -3  -2   0   1  -9  -8  -6
    7 | 0  1   2   3  -3  -2  -1   0  -9  -8  -7
    8 | 0  0   0   0   9   9   9   9   0   0   0
    9 | 0  0  -1  -1   9   9   8   8   0   0  -1
   10 | 0  1   0   1   6   7   6   7   0   1   0

Prog

(PARI) A(n, k) = { my (v = 0, t = 1); while (n && k, v += (bittest(n, 1)*bittest(k, 0) - bittest(n, 0)*bittest(k, 1)) * t; n \= 2; k \= 2; t *= 3; ); return (v); }

Crossrefs

See A372345 for a similar sequence.

Sequence in context: A247706 A361527 A247704 * A127802 A165951 A300288

Adjacent sequences: A372341 A372342 A372343 * A372345 A372346 A372347

Keyword

sign,base,tabl

Author

Rémy Sigrist, Apr 28 2024

Status

approved