A166556 Triangle read by rows, A000012 * A047999.

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

Offset: 0

Author

Gary W. Adamson, Oct 17 2009

Keywords

• nonn
• easy
• tabl

Examples

			First few rows of the triangle =
   1;
   2, 1;
   3, 1, 1;
   4, 2, 2, 1;
   5, 2, 2, 1, 1;
   6, 3, 2, 1, 2, 1;
   7, 3, 3, 1, 3, 1, 1;
   8, 4, 4, 2, 4, 2, 2, 1;
   9, 4, 4, 2, 4, 2, 2, 1, 1;
  10, 5, 4, 2, 4, 2, 2, 1, 2, 1;
  11, 5, 5, 2, 4, 2, 2, 1, 3, 1, 1;
  12, 6, 6, 3, 4, 2, 2, 1, 4, 2, 2, 1;
  13, 6, 6, 3, 5, 2, 2, 1, 5, 2, 2, 1, 1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A000027, A004524, A004526, A047999, A080100.

Sums include: A006046 (row), A007729 (diagonal).

Programs

Magma

A166556:= func< n,k | (&+[(Binomial(j,k) mod 2): j in [k..n]]) >;
[A166556(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 02 2024

Maple

A166556 := proc(n,k)
    local j;
    add(A047999(j,k),j=k..n) ;
end proc: # R. J. Mathar, Jul 21 2016

Mathematica

A166556[n_, k_]:= Sum[Mod[Binomial[j,k], 2], {j,k,n}];
Table[A166556[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2024 *)

Python

def A166556(n,k): return sum(binomial(j,k)%2 for j in range(k,n+1))
print(flatten([[A166556(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 02 2024

Formula

Triangle read by rows, A000012 * A047999; where A000012 = an infinite lower triangular matrix with all 1's: [1; 1,1; 1,1,1;..]; and A047999 = Sierpinski's gasket.

The operation takes partial sums of Sierpinski's gasket terms, by columns.

From G. C. Greubel, Dec 02 2024: (Start)

T(n, k) = Sum_{j=k..n} (binomial(j,k) mod 2).

T(n, 0) = A000027(n+1).

T(n, 1) = A004526(n+1).

T(n, 2) = A004524(n+1).

T(2*n, n) = A080100(n).

Sum_{k=0..n} T(n, k) = A006046(n+1).

Sum_{k=0..n} (-1)^k*T(n, k) = A006046(floor(n/2)+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A007729(n). (End)