OFFSET
1,3
LINKS
G. C. Greubel, Rows n = 1..100 of the triangle, flattened
FORMULA
Odd rows: n terms of n, 0, n, ...; even rows, n terms of 0, n, 0, ...
T(n,k) = n if n+k even, T(n,k) = 0 if n+k odd.
Sum_{k=1..n} T(n, k) = A093005(n) (row sums).
From G. C. Greubel, Mar 13 2024: (Start)
T(n, k) = n*(1 + (-1)^(n+k))/2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n+1)*A093005(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n) * A000326(floor((n+1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*A123684(floor((n+1)/2)). (End)
EXAMPLE
First few rows of the triangle:
1;
0, 2;
3, 0, 3;
0, 4, 0, 4;
5, 0, 5, 0, 5;
...
MATHEMATICA
Table[n*(1+(-1)^(n+k))/2, {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Mar 13 2024 *)
PROG
(Magma) [n*(1+(-1)^(n+k))/2: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 13 2024
(SageMath) flatten([[n*(1+(-1)^(n+k))//2 for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 13 2024
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Mar 14 2007
EXTENSIONS
More terms added by G. C. Greubel, Mar 13 2024
STATUS
approved