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

Author

Roger L. Bagula, Sep 27 2007

Keywords

• nonn
• tabl
• less
• changed

Examples

			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;
  ...
		

Links

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

Crossrefs

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

Programs

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

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 *)

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

Formula

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

Extensions

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