cp's OEIS Frontend

A154957 A symmetric (0,1)-triangle.

%I A154957 #17 Aug 16 2025 07:04:52
%S A154957 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,
%T A154957 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,
%U A154957 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
%N A154957 A symmetric (0,1)-triangle.
%C A154957 Parity of A003983. - _Jeremy Gardiner_, Mar 09 2014
%H A154957 G. C. Greubel, <a href="/A154957/b154957.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A154957 T(n,k) = Sum_{j=0..n} [j<=k]*[j<=n-k]*(mod(j+1,2) - mod(j,2)).
%F A154957 T(2*n, n) - T(2*n, n+1) = (-1)^n.
%F A154957 T(2*n, n) = (n+1) mod 2.
%F A154957 Sum_{k=0..n} T(n, k) = A004524(n+3).
%F A154957 Sum_{k=0..floor(n/2)} T(n-k, k) = A154958(n) (diagonal sums).
%F A154957 From _G. C. Greubel_, Mar 07 2022: (Start)
%F A154957 T(n, n-k) = T(n, k).
%F A154957 Sum_{k=0..floor(n/2)} T(n, k) = floor((n+4)/4).
%F A154957 T(2*n+1, n) = (1+(-1)^n)/2. (End)
%e A154957 Triangle begins
%e A154957   1;
%e A154957   1, 1;
%e A154957   1, 0, 1;
%e A154957   1, 0, 0, 1;
%e A154957   1, 0, 1, 0, 1;
%e A154957   1, 0, 1, 1, 0, 1;
%e A154957   1, 0, 1, 0, 1, 0, 1;
%e A154957   1, 0, 1, 0, 0, 1, 0, 1;
%e A154957   1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A154957   1, 0, 1, 0, 1, 1, 0, 1, 0, 1;
%e A154957   1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A154957   1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1;
%e A154957   1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A154957   1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1;
%e A154957   1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1;
%e A154957   1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1;
%t A154957 T[n_, k_]:= Sum[(Mod[j+1,2] - Mod[j,2]), {j,0,Min[k,n-k]}];
%t A154957 Table[T[n, k], {n,0,20}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 07 2022 *)
%o A154957 (Sage)
%o A154957 def A154957(n,k): return sum( (j+1)%2 - j%2 for j in (0..min(k,n-k)) )
%o A154957 flatten([[A154957(n,k) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Mar 07 2022
%Y A154957 Cf. A003983, A004524 (row sums), A154958 (diagonal sums), A158856.
%K A154957 easy,nonn,tabl
%O A154957 0,1
%A A154957 _Paul Barry_, Jan 18 2009