A154957 A symmetric (0,1)-triangle.

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

Offset: 0

Author

Paul Barry, Jan 18 2009

Keywords

• easy
• nonn
• tabl

Comments

Parity of A003983. - Jeremy Gardiner, Mar 09 2014

Examples

			Triangle begins
  1;
  1, 1;
  1, 0, 1;
  1, 0, 0, 1;
  1, 0, 1, 0, 1;
  1, 0, 1, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A003983, A004524 (row sums), A154958 (diagonal sums), A158856.

Programs

Mathematica

T[n_, k_]:= Sum[(Mod[j+1,2] - Mod[j,2]), {j,0,Min[k,n-k]}];
Table[T[n, k], {n,0,20}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 07 2022 *)

Sage

def A154957(n,k): return sum( (j+1)%2 - j%2 for j in (0..min(k,n-k)) )
flatten([[A154957(n,k) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Mar 07 2022

Formula

T(n,k) = Sum_{j=0..n} [j<=k]*[j<=n-k]*(mod(j+1,2) - mod(j,2)).

T(2*n, n) - T(2*n, n+1) = (-1)^n.

T(2*n, n) = (n+1) mod 2.

Sum_{k=0..n} T(n, k) = A004524(n+3).

Sum_{k=0..floor(n/2)} T(n-k, k) = A154958(n) (diagonal sums).

From G. C. Greubel, Mar 07 2022: (Start)

T(n, n-k) = T(n, k).

Sum_{k=0..floor(n/2)} T(n, k) = floor((n+4)/4).

T(2*n+1, n) = (1+(-1)^n)/2. (End)