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A082110
Array A(n,k) = (k*n)^2 + 5*(k*n) + 1, read by antidiagonals.
6
1, 1, 1, 1, 7, 1, 1, 15, 15, 1, 1, 25, 37, 25, 1, 1, 37, 67, 67, 37, 1, 1, 51, 105, 127, 105, 51, 1, 1, 67, 151, 205, 205, 151, 67, 1, 1, 85, 205, 301, 337, 301, 205, 85, 1, 1, 105, 267, 415, 501, 501, 415, 267, 105, 1, 1, 127, 337, 547, 697, 751, 697, 547, 337, 127, 1
OFFSET
0,5
LINKS
FORMULA
A(n, k) = (k*n)^2 + 5*(k*n) + 1 (Square array).
A(k, n) = A(n, k).
A(2, k) = A082111(k).
A(3, k) = A082112(k).
A(n, n) = T(2*n, n) = A082113(n) (main diagonal).
T(n, k) = (k*(n-k))^2 + 5*k*(n-k) + 1 (number triangle).
Sum_{k=0..n} T(n, k) = A082114(n) (diagonal sums of the array).
From G. C. Greubel, Dec 22 2022: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} (-1)^k*T(n, k) = (1 - 3*n)*(1 + (-1)^n)/2. (End)
EXAMPLE
Square array, A(n, k), begins as:
1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012;
1, 7, 15, 25, 37, 51, 67, 85, 105, ... A082111;
1, 15, 37, 67, 105, 151, 205, 267, 337, ... A082112;
1, 25, 67, 127, 205, 301, 415, 547, 697, ...
1, 37, 105, 205, 337, 501, 697, 925, 1185, ...
1, 51, 151, 301, 501, 751, 1051, 1401, 1801, ...
1, 67, 205, 415, 697, 1051, 1477, 1975, 2545, ...
1, 85, 267, 547, 925, 1401, 1975, 2647, 3417, ...
1, 105, 337, 697, 1185, 1801, 2545, 3417, 4417, ...
Antidiagonals, T(n, k), begins as:
1;
1, 1;
1, 7, 1;
1, 15, 15, 1;
1, 25, 37, 25, 1;
1, 37, 67, 67, 37, 1;
1, 51, 105, 127, 105, 51, 1;
1, 67, 151, 205, 205, 151, 67, 1;
1, 85, 205, 301, 337, 301, 205, 85, 1;
MATHEMATICA
T[n_, k_]:= (k*(n-k))^2 + 5*(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 + 5*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022
(SageMath)
def A082110(n, k): return (k*(n-k))^2 + 5*(k*(n-k)) + 1
flatten([[A082110(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, Apr 04 2003
STATUS
approved