OFFSET
1,2
COMMENTS
A(n,k) is also the distance from A(n, k-1) to the earliest square greater than 3*A(n,k-1) - A(n,k-2).
In column k, every term is the arithmetic mean of its neighbors minus A000217(k).
LINKS
G. C. Greubel, Antidiagonals n = 1..50, flattened
FORMULA
A(n,k) = A000217(k)*n^2 + k*n + 1, for k odd.
A(n,k) = A000217(k)*n^2 + (k+1)*n = (k+1)*x*(k*n/2 + 1), for k even.
A(n,k) = (A(n,k-1) + A(n,k+1) + k*(k+1))/2, for any k.
A(n, 0) = A000027(n).
A(n, 1) = A002061(n+1).
A(n, 2) = A028896(n).
A(n, 3) = A085473(n).
From G. C. Greubel, Dec 25 2022: (Start)
A(n, k) = (1/2)*( (k*n+1)*(k*n+n+1) + (-1)^k*(n-1) ).
T(n, k) = (1/2)*( (k*(n-k)+1)*((k+1)*(n-k)+1) + (-1)^k*(n-k-1) ).
Sum_{k=0..n-1} T(n, k) = (1/120)*(2*n^5 + 5*n^4 + 20*n^3 + 25*n^2 + 98*n - 15*(1-(-1)^n)). (End)
EXAMPLE
Array, A(n, k), begins:
1 3 6 10 15 21 28 36 45 ... A000217;
2 7 18 31 50 71 98 127 162 ... A195605;
3 13 36 64 105 151 210 274 351 ...
4 21 60 109 180 261 364 477 612 ...
5 31 90 166 275 401 560 736 945 ...
6 43 126 235 390 571 798 1051 1350 ...
7 57 168 316 525 771 1078 1422 1827 ...
8 73 216 409 680 1001 1400 1849 2376 ...
9 91 270 514 855 1261 1764 2332 2997 ...
Antidiagonals, T(n, k), begin as:
1;
2, 3;
3, 7, 6;
4, 13, 18, 10;
5, 21, 36, 31, 15;
6, 31, 60, 64, 50, 21;
7, 43, 90, 109, 105, 71, 28;
8, 57, 126, 166, 180, 151, 98, 36;
9, 73, 168, 235, 275, 261, 210, 127, 45;
10, 91, 216, 316, 390, 401, 364, 274, 162, 55;
MATHEMATICA
A[n_, 0]:= n; A[n_, k_]:= (k*n+1)^2 -A[n, k-1]; Table[Function[n, A[n, k]][m-k+1], {m, 0, 10}, {k, 0, m}]//Flatten (* Michael De Vlieger, Oct 27 2021 *)
PROG
(Magma)
A347533:= func< n, k | (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1)) >;
[A347533(n, k): k in [0..n-1], n in [1..13]]; // G. C. Greubel, Dec 25 2022
(SageMath)
def A347533(n, k): return (1/2)*((k*(n-k)+1)*((k+1)*(n-k)+1) +(-1)^k*(n-k- 1))
flatten([[A347533(n, k) for k in range(n)] for n in range(1, 14)]) # G. C. Greubel, Dec 25 2022
CROSSREFS
KEYWORD
AUTHOR
Lamine Ngom, Sep 05 2021
STATUS
approved