A082105 Array A(n, k) = (k*n)^2 + 4*(k*n) + 1, read by antidiagonals.
1, 1, 1, 1, 6, 1, 1, 13, 13, 1, 1, 22, 33, 22, 1, 1, 33, 61, 61, 33, 1, 1, 46, 97, 118, 97, 46, 1, 1, 61, 141, 193, 193, 141, 61, 1, 1, 78, 193, 286, 321, 286, 193, 78, 1, 1, 97, 253, 397, 481, 481, 397, 253, 97, 1, 1, 118, 321, 526, 673, 726, 673, 526, 321, 118, 1
Offset: 0
Examples
Array, A(n, k), begins as: 1, 1, 1, 1, 1, 1, ... A000012; 1, 6, 13, 22, 33, 46, ... A028872; 1, 13, 33, 61, 97, 141, ... A082109; 1, 22, 61, 118, 193, 286, ... ; 1, 33, 97, 193, 321, 481, ... ; 1, 46, 141, 286, 481, 726, ... ; Triangle, T(n, k), begins as: 1; 1, 1; 1, 6, 1; 1, 13, 13, 1; 1, 22, 33, 22, 1; 1, 33, 61, 61, 33, 1; 1, 46, 97, 118, 97, 46, 1; 1, 61, 141, 193, 193, 141, 61, 1; 1, 78, 193, 286, 321, 286, 193, 78, 1;
Links
- G. C. Greubel, Antidiagonals n = 0..50, flattened
Programs
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Magma
[(k*(n-k))^2 + 4*(k*(n-k)) + 1: k in [0..n], n in [0..13]]; // G. C. Greubel, Dec 22 2022
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Mathematica
T[n_, k_]:= (k*(n-k))^2 + 4*(k*(n-k)) + 1; Table[T[n,k], {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2022 *) -
SageMath
def A082105(n,k): return (k*(n-k))^2 + 4*(k*(n-k)) + 1 flatten([[A082105(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 22 2022
Formula
A(n, k) = (k*n)^2 + 4*(k*n) + 1 (square array).
A(n, n) = T(2*n, n) = A082106(n) (main diagonal).
T(n, k) = A(n-k, k) (number triangle).
Sum_{k=0..n} T(n, k) = A082107(n) (diagonal sums).
T(n, n-1) = A028872(n-1), n >= 1.
T(n, n-2) = A082109(n-2), n >= 2.
From G. C. Greubel, Dec 22 2022: (Start)
Sum_{k=0..n} (-1)^k * T(n, k) = ((1+(-1)^n)/2)*A016897(n-1).
T(2*n+1, n+1) = A047673(n+1), n >= 0.
T(n, n-k) = T(n, k). (End)
Comments