OFFSET
0,5
LINKS
G. C. Greubel, Antidiagonals n = 0..50, flattened
FORMULA
A(n, k) = (k*n)^2 + 3*k*n + 1 (square array).
A(k, n) = A(n, k).
A(n, n) = T(2*n, n) = A057721(n).
A(n, n+1) = A072025(n).
T(n, k) = (k*(n-k))^2 + 3*k*(n-k) + 1 (antidiagonals).
Sum_{k=0..n} T(n, k) = A082047(n) (antidiagonal sums).
From G. C. Greubel, Dec 22 2022: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1 + (-1)^n)*(1 - 2*n).
T(2*n+1, n-1) = T(2*n-1, n-1) = A072025(n-1). (End)
EXAMPLE
Array, A(n, k), begins as:
1, 1, 1, 1, 1, 1, 1, 1, ... A000012;
1, 5, 11, 19, 29, 41, 55, 71, ... A028387;
1, 11, 29, 55, 89, 131, 181, 239, ... A082108;
1, 19, 55, 109, 181, 271, 379, 505, ... A069131;
1, 29, 89, 181, 305, 461, 649, 869, ... ;
1, 41, 131, 271, 461, 701, 991, 1331, ... ;
1, 55, 181, 379, 649, 991, 1405, 1891, ... ;
1, 71, 239, 505, 869, 1331, 1891, 2549, ... ;
Antidiagonals, T(n, k), begin as:
1;
1, 1;
1, 5, 1;
1, 11, 11, 1;
1, 19, 29, 19, 1;
1, 29, 55, 55, 29, 1;
1, 41, 89, 109, 89, 41, 1;
1, 55, 131, 181, 181, 131, 55, 1;
1, 71, 181, 271, 305, 271, 181, 71, 1;
MATHEMATICA
T[n_, k_]:= (k*(n-k))^2 + 3*(k*(n-k)) + 1;
Table[T[n, k], {n, 0, 13}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 22 2022 *)
PROG
(Magma) [(k*(n-k))^2 + 3*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022
(SageMath)
def A082046(n, k): return (k*(n-k))^2 + 3*(k*(n-k)) + 1
flatten([[A082046(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Apr 03 2003
STATUS
approved