A140820 Triangle read by rows: T(n,k) = 1 if and only if the Gray codes for n and k have no bits in common.

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

Offset

1,1

Comments

Old title was: "Let c(j, k) = (floor(j/2^k) mod 2) and b(j, k) = if( c(j, k) = 0 and c(j, k+1) = 0 then 0 else if(c(j, k) = 1 and c(j, k+1) = 1 then 0 else 1)) then the triangle is generated by T(n, k) = if( Sum_{j=0..n} b(n, j)* b(k, j) = 0 then 1 else 0)."

Row sums are {1, 1, 1, 2, 2, 1, 2, 4, 4, 2, 1, 2, 4, 2, 4, 8, 8, ...}.

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Formula

Let c(j, k) = (floor(j/2^k) mod 2) and b(j, k) = if( c(j, k) = 0 and c(j, k+1) = 0 then 0 else if(c(j, k) = 1 and c(j, k+1) = 1 then 0 else 1)) then the triangle is generated by T(n, k) = if( Sum_{j=0..n} b(n, j)* b(k, j) = 0 then 1 else 0).

Example

Triangle begins as:
  1;
  1, 0;
  1, 0, 0;
  1, 1, 0, 0;
  1, 1, 0, 0, 0;
  1, 0, 0, 0, 0, 0;
  1, 0, 0, 1, 0, 0, 0;
  1, 1, 1, 1, 0, 0, 0, 0;
  1, 1, 1, 1, 0, 0, 0, 0, 0;
  1, 0, 0, 1, 0, 0, 0, 0, 0, 0;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;
  1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

Mathematica

n=16; c[i_, k_]:= Mod[Floor[i/2^k], 2]; b[i_, k_]:= If[c[i, k]==0 && c[i, k+1]==0, 0, If[c[i, k]==1 && c[i, k+1]==1, 0, 1]]; Table[If[Sum[b[i, k]* b[j, k], {k, 0, n}]==0, 1, 0], {i, 0, n}, {j, 0, i} ]//Flatten (* modified by G. C. Greubel, May 30 2019 *)

Prog

(PARI)
c(j, k) = Mod((j/2^k)\1, 2);
b(j, k) = if(c(j, k)==0 && c(j, k+1)==0, 0, if(c(j, k)==1 && c(j, k+1)==1, 0, 1));
for(k=0, 16, for(s=0, k, print1(if(sum(r=0, k, b(k, r)*b(s, r))==0, 1, 0), ", "))) \\ G. C. Greubel, Jun 03 2019
(PARI) T(n, k) = !bitand(bitxor(n, n>>1), bitxor(k, k>>1)); \\ Kevin Ryde, Jul 13 2020
(Sage)
def c(j, k): return Mod(floor(j/2^k), 2)
def b(j, k):
    if (c(j, k)==0 and c(j, k+1)==0): return 0
    elif (c(j, k)==1 and c(j, k+1)==1): return 0
    else: return 1
def T(n, k):
    if (sum(b(n, r)*b(k, r) for r in (0..n))==0): return 1
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..16)] # G. C. Greubel, Jun 03 2019

Crossrefs

Cf. A131218, A014550.

Sequence in context: A176918 A176890 A164057 * A275661 A266716 A190242

Adjacent sequences: A140817 A140818 A140819 * A140821 A140822 A140823

Keyword

nonn,tabl,less

Author

Roger L. Bagula and Gary W. Adamson, Oct 17 2008

Extensions

Edited by G. C. Greubel, May 30 2019

New title from Charlie Neder, Jun 03 2019

Status

approved