A332351 Triangle read by rows: T(m,n) = Sum_{-m<i<m, -n<j<n, gcd{i,j}=1} (m-|i|)*(n-|j|)/2, m >= n >= 1.

0, 1, 6, 2, 13, 28, 3, 22, 49, 86, 4, 33, 74, 131, 200, 5, 46, 105, 188, 289, 418, 6, 61, 140, 251, 386, 559, 748, 7, 78, 181, 326, 503, 730, 979, 1282, 8, 97, 226, 409, 632, 919, 1234, 1617, 2040, 9, 118, 277, 502, 777, 1132, 1521, 1994, 2517, 3106, 10, 141, 332, 603, 934, 1361, 1828, 2397, 3026, 3735, 4492

Offset

1,3

Comments

This is the triangle in A332350, halved.

This triangle is the lower half of the array defined in A115009.

References

Jovisa Zunic, Note on the number of two-dimensional threshold functions, SIAM J. Discrete Math. Vol. 25 (2011), No. 3, pp. 1266-1268. See Equation (1.2).

Links

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

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)/2.

Example

Triangle begins:
0,
1, 6,
2, 13, 28,
3, 22, 49, 86,
4, 33, 74, 131, 200,
5, 46, 105, 188, 289, 418,
6, 61, 140, 251, 386, 559, 748,
7, 78, 181, 326, 503, 730, 979, 1282,
8, 97, 226, 409, 632, 919, 1234, 1617, 2040,
9, 118, 277, 502, 777, 1132, 1521, 1994, 2517, 3106,
...

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)/2, n=1..m), ); od:

Mathematica

A332351[m_, n_]:=Sum[If[CoprimeQ[i, j], 2(m-i)(n-j), 0], {i, m-1}, {j, n-1}]+2m*n-m-n; Table[A332351[m, n], {m, 15}, {n, m}] (* Paolo Xausa, Oct 18 2023 *)

Crossrefs

The main diagonal is A141255, or A114043 - 1.

Cf. A115009, A332350, A332352.

This is the lower triangle of the array in A115009.

Sequence in context: A106034 A194036 A194100 * A142867 A080977 A095951

Adjacent sequences: A332348 A332349 A332350 * A332352 A332353 A332354

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 10 2020

Status

approved