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

0,1

Comments

Parity of A003983. - Jeremy Gardiner, Mar 09 2014

Links

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

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:

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)

Example

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;

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A131219 A127970 A158856 * A140865 A114000 A131218

Adjacent sequences: A154954 A154955 A154956 * A154958 A154959 A154960

Keyword

easy,nonn,tabl

Author

Paul Barry, Jan 18 2009

Status

approved