A386456 Square array read by ascending antidiagonals, where row n lists the parity of the n-th differences of the partition numbers (A000041).

1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1

Offset

0

Links

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Paolo Xausa, Bitmap image showing the NW corner of the array (400 X 400 values, where white = 0 and black = 1).

Formula

A(0,k) = A000041(k) mod 2 = A040051(k).

A(n,0) = A196251(n).

A(n,n) = A379378(n) mod 2 = A386457(n).

A(n,k) = |A(n-1,k+1) - A(n-1,k)| = A(n-1,k+1) XOR A(n-1,k), for n >= 1.

A(n,k) = A175804(k,n) mod 2.

Example

The array begins:
  n\k| 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15  ...
  --------------------------------------------------------
   0 | 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, ... = A040051
   1 | 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, ...
   2 | 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, ...
   3 | 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, ...
   4 | 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, ...
   5 | 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, ...
   6 | 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, ...
   7 | 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, ...
   8 | 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, ...
   9 | 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, ...
  10 | 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, ...
  11 | 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, ...
  12 | 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, ...
  13 | 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, ...
  14 | 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, ...
  15 | 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, ...
  ...  |                                             \______ A386457 (main diagonal)
    A196251

Mathematica

A386456list[dmax_] := Table[Diagonal[#, d], {d, 1 - dmax, 0}] & [Reverse[NestList[Abs[Differences[#]] &, Mod[PartitionsP[Range[0, dmax - 1]], 2], dmax - 1]]];
A386456list[15] (* Generates 15 antidiagonals *)

Crossrefs

Cf. A000041, A040051 (first row), A175804, A196251 (first column), A386457 (main diagonal), A386458 (antidiagonal sums).

Sequence in context: A089045 A259599 A070749 * A285341 A059778 A355690

Adjacent sequences: A386453 A386454 A386455 * A386457 A386458 A386459

Keyword

nonn,tabl

Author

Paolo Xausa, Jul 22 2025

Status

approved