A319047 Square array A(n,k) where column k is balanced (2k+1)-ary enumeration of integers; n>=0, k>=1, read by antidiagonals.

0, 0, 1, 0, 1, -1, 0, 1, 2, 3, 0, 1, 2, -2, 4, 0, 1, 2, 3, -1, 2, 0, 1, 2, 3, -3, 5, -3, 0, 1, 2, 3, 4, -2, 6, -2, 0, 1, 2, 3, 4, -4, -1, 7, -4, 0, 1, 2, 3, 4, 5, -3, 7, 3, 9, 0, 1, 2, 3, 4, 5, -5, -2, 8, 4, 10, 0, 1, 2, 3, 4, 5, 6, -4, -1, 9, 10, 8, 0, 1, 2, 3, 4, 5, 6, -6, -3, 9, 10, 11, 12

Offset: 0

Author

Alois P. Heinz, Aug 26 2019

Keywords

• sign
• look
• hear
• tabl

Examples

			Square array A(n,k) begins:
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
   1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  -1,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
   3, -2,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
   4, -1, -3,  4,  4,  4,  4,  4,  4,  4,  4, ...
   2,  5, -2, -4,  5,  5,  5,  5,  5,  5,  5, ...
  -3,  6, -1, -3, -5,  6,  6,  6,  6,  6,  6, ...
  -2,  7,  7, -2, -4, -6,  7,  7,  7,  7,  7, ...
  -4,  3,  8, -1, -3, -5, -7,  8,  8,  8,  8, ...
   9,  4,  9,  9, -2, -4, -6, -8,  9,  9,  9, ...
  10, 10, 10, 10, -1, -3, -5, -7, -9, 10, 10, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..200, flattened
• Wikipedia, Balanced ternary

Crossrefs

Columns k=1-4 give: A117966, A309991, A309995, A316823.

A(n,n+1) gives A001477.

A(n+1,n) gives A001478 (for n>0).

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 0,
      (b-> b*A(iquo(n, b), k)+mods(n, b))(2*k+1))
    end:
seq(seq(A(n, 1+d-n), n=0..d), d=0..14);

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 0, With[{b = 2k+1},
     b*A[Quotient[n, b], k] + Mod[n, b, -Quotient[b-1, 2]]]];
Table[Table[A[n, 1+d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 09 2021, after Alois P. Heinz *)