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Triangle read by rows: T(n,k) = (n + k)/2 if (n + k) is congruent to 0 (mod 2), otherwise T(n,k) = 0; n >= 1, k >= 1.
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%I #14 Jul 21 2022 03:11:02

%S 1,0,2,2,0,3,0,3,0,4,3,0,4,0,5,0,4,0,5,0,6,4,0,5,0,6,0,7,0,5,0,6,0,7,

%T 0,8,5,0,6,0,7,0,8,0,9,0,6,0,7,0,8,0,9,0,10,6,0,7,0,8,0,9,0,10,0,11,0,

%U 7,0,8,0,9,0,10,0,11,0,12,7,0,8,0,9,0,10,0,11,0,12,0,13,0

%N Triangle read by rows: T(n,k) = (n + k)/2 if (n + k) is congruent to 0 (mod 2), otherwise T(n,k) = 0; n >= 1, k >= 1.

%C Row sums see A001318.

%e The triangle begins:

%e k=1 2 3 4 5 6

%e n=1: 1;

%e n=2: 0, 2;

%e n=3: 2, 0, 3;

%e n=4: 0, 3, 0, 4;

%e n=5: 3, 0, 4, 0, 5;

%e n=6: 0, 4, 0, 5, 0, 6;

%e and so on.

%t T[n_, k_] := If[EvenQ[n + k], (n + k)/2, 0]; Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Jul 19 2022 *)

%Y Cf. A001318, A138099 (without the zeros).

%K nonn,tabl

%O 1,3

%A _Ctibor O. Zizka_, Jul 19 2022