A332350 Triangle read by rows: T(m,n) = Sum_{-m<i<m, -n<j<n, gcd{i,j}=1} (m-|i|)*(n-|j|), m >= n >= 1.

0, 2, 12, 4, 26, 56, 6, 44, 98, 172, 8, 66, 148, 262, 400, 10, 92, 210, 376, 578, 836, 12, 122, 280, 502, 772, 1118, 1496, 14, 156, 362, 652, 1006, 1460, 1958, 2564, 16, 194, 452, 818, 1264, 1838, 2468, 3234, 4080, 18, 236, 554, 1004, 1554, 2264, 3042, 3988, 5034, 6212

Offset

1,2

Links

Table of n, a(n) for n=1..55.

Max A. Alekseyev, On the number of two-dimensional threshold functions, arXiv:math/0602511 [math.CO], 2006-2010; SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631. doi:10.1137/090750184. See N(m,n) in Theorem 2.

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. This sequence is f_1(m,n).

Example

Triangle begins:
0,
2, 12,
4, 26, 56,
6, 44, 98, 172,
8, 66, 148, 262, 400,
10, 92, 210, 376, 578, 836,
12, 122, 280, 502, 772, 1118, 1496,
14, 156, 362, 652, 1006, 1460, 1958, 2564,
16, 194, 452, 818, 1264, 1838, 2468, 3234, 4080,
18, 236, 554, 1004, 1554, 2264, 3042, 3988, 5034, 6212,
...

Maple

VR := proc(m, n, q) local a, i, j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i, j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
for m from 1 to 12 do lprint(seq(VR(m, n, 1), n=1..m), ); od:

Mathematica

T[m_, n_] := Sum[Boole[GCD[i, j] == 1] (m - Abs[i]) (n - Abs[j]), {i, -m + 1, m - 1}, {j, -n + 1, n - 1}];
Table[T[m, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Apr 19 2020 *)

Crossrefs

The main diagonal is A331771.

Cf. A332351, A332352.

Sequence in context: A248030 A082292 A248588 * A164857 A326125 A066700

Adjacent sequences: A332347 A332348 A332349 * A332351 A332352 A332353

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 10 2020

Status

approved