A131218 Array read by antidiagonals: A(n, k) = 1 if and only if the Gray codes for n and k have no bits in common.

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

Offset

0,1

Links

G. C. Greubel, Antidiagonals n = 0..100, flattened

Formula

A(n,k) = A(k,n) = A140820(n,k) for k <= n.

Example

Array, A(n, k), begins as:
  1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
  1, 0, 0, 1, 1, 0, 0, 1, 1, ...;
  1, 0, 0, 0, 0, 0, 0, 1, 1, ...;
  1, 1, 0, 0, 0, 0, 1, 1, 1, ...;
  1, 1, 0, 0, 0, 0, 0, 0, 0, ...;
  1, 0, 0, 0, 0, 0, 0, 0, 0, ...;
  1, 0, 0, 1, 0, 0, 0, 0, 0, ...;
  1, 1, 1, 1, 0, 0, 0, 0, 0, ...;
  1, 1, 1, 1, 0, 0, 0, 0, 0, ...;
  ...
Antidiagonals begin as:
  1;
  1, 1;
  1, 0, 1;
  1, 0, 0, 1;
  1, 1, 0, 1, 1;
  1, 1, 0, 0, 1, 1;
  1, 0, 0, 0, 0, 0, 1;
  1, 0, 0, 0, 0, 0, 0, 1;
  1, 1, 0, 0, 0, 0, 0, 1, 1;
  1, 1, 1, 1, 0, 0, 1, 1, 1, 1;
  1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1;
  1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1;
  1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Mathematica

A131218[n_, k_]:= Boole[BitAnd[BitXor[n, BitShiftRight[n, 1]], BitXor[k, BitShiftRight[k, 1]]]==0]; (* based on Kevin Ryde's code of A140820 *)
Table[A131218[n-k, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 04 2025 *)

Prog

(Magma)
A131218:= func< n, k | BitwiseAnd(BitwiseXor(n, ShiftRight(n, 1)), BitwiseXor(k, ShiftRight(k, 1))) eq 0 select 1 else 0 >; // based on Kevin Ryde's code of A140820
[A131218(n-k, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Sep 04 2025
(PARI) A(n, k) = !bitand(bitxor(n, n>>1), bitxor(k, k>>1)); \\ Joerg Arndt, Sep 05 2025
(SageMath)
def A131218(n, k): return int( (n^^(n>>1)) & (k^^(k>>1)) ==0) # based on Kevin Ryde's code of A140820
print(flatten([[A131218(n-k, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Sep 05 2025

Crossrefs

Cf. A140820 (lower triangle), A363710 (antidiagonal sums).

Sequence in context: A154957 A140865 A114000 * A174391 A343910 A329682

Adjacent sequences: A131215 A131216 A131217 * A131219 A131220 A131221

Keyword

nonn,tabl,less

Author

Roger L. Bagula, Sep 27 2007

Extensions

Edited by and new name from G. C. Greubel, Sep 04 2025

Status

approved