OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, m) = (n + 1)*(if m <= floor(n/2) then (m - 1)*(m + 2) / 2 else (n - m + 2)*(n - (m + 1)) / 2 fi) + n. - Georg Fischer, Oct 28 2023
From G. C. Greubel, Jun 10 2024: (Start)
T(n, k) = n + (n+1)*(k-1)*(k+2)/2 if 0 <= k <= floor(n/2), otherwise T(n, k) = T(n, n-k).
Sum_{k=0..n} T(n, k) = (1/48)*(n+1)*(-53 - 5*n + 3*(-1)^n*(n+1) + 2*(n + 1)^3). (End)
EXAMPLE
Triangle begins as:
-1;
-1, -1;
-1, 2, -1;
-1, 3, 3, -1;
-1, 4, 14, 4, -1;
-1, 5, 17, 17, 5, -1;
-1, 6, 20, 41, 20, 6, -1;
-1, 7, 23, 47, 47, 23, 7, -1;
-1, 8, 26, 53, 89, 53, 26, 8, -1;
-1, 9, 29, 59, 99, 99, 59, 29, 9, -1;
-1, 10, 32, 65, 109, 164, 109, 65, 32, 10, -1;
MAPLE
seq(print(seq((n + 1) * (if m <= n/2 then (m - 1) * (m + 2)\
/ 2 else (n - m + 2) * (n - (m + 1)) / 2 fi) + n, m=0..n)), n=0..10); # Georg Fischer, Oct 28 2023
MATHEMATICA
T[n_, k_]:= If[k<=Floor[n/2], n +(n+1)*(k-1)*(k+2)/2, T[n, n-k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
PROG
(Magma)
function T(n, k) // A143199
if k le Floor(n/2) then return n + (n+1)*(k-1)*(k+2)/2;
else return T(n, n-k);
end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 10 2024
(SageMath)
flatten([[A143199(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 10 2024
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula and Gary W. Adamson, Oct 20 2008
EXTENSIONS
Definition clarified and offset corrected by Georg Fischer, Oct 28 2023
STATUS
approved