A332350
Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.
Original entry on oeis.org
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
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,
...
- 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).
-
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:
-
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 *)
A332352
Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.
Original entry on oeis.org
0, 0, 0, 2, 4, 16, 4, 8, 28, 48, 6, 12, 44, 76, 120, 8, 16, 60, 104, 164, 224, 10, 20, 80, 140, 224, 308, 424, 12, 24, 100, 176, 284, 392, 540, 688, 14, 28, 124, 220, 356, 492, 680, 868, 1096, 16, 32, 148, 264, 428, 592, 820, 1048, 1324, 1600, 18, 36, 176, 316, 516, 716, 996, 1276, 1616, 1956, 2392
Offset: 1
Triangle begins:
0,
0, 0,
2, 4, 16,
4, 8, 28, 48,
6, 12, 44, 76, 120,
8, 16, 60, 104, 164, 224,
10, 20, 80, 140, 224, 308, 424,
12, 24, 100, 176, 284, 392, 540, 688,
14, 28, 124, 220, 356, 492, 680, 868, 1096,
16, 32, 148, 264, 428, 592, 820, 1048, 1324, 1600,
...
-
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,2),n=1..m),); od:
-
A332352[m_,n_]:=Sum[If[GCD[i,j]==2,4(m-i)(n-j),0],{i,2,m-1,2},{j,2,n-1,2}]+If[n>2,2(m*n-2m),0]+If[m>2,2(m*n-2n),0];Table[A332352[m, n],{m,15},{n, m}] (* Paolo Xausa, Oct 18 2023 *)
A115009
Array read by antidiagonals: let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)*(n+1-j), then T(m,n) = 2*m*n+m+n+2*V(m,n), for m >= 0, n >= 0.
Original entry on oeis.org
0, 1, 1, 2, 6, 2, 3, 13, 13, 3, 4, 22, 28, 22, 4, 5, 33, 49, 49, 33, 5, 6, 46, 74, 86, 74, 46, 6, 7, 61, 105, 131, 131, 105, 61, 7, 8, 78, 140, 188, 200, 188, 140, 78, 8, 9, 97, 181, 251, 289, 289, 251, 181, 97, 9, 10, 118, 226, 326, 386, 418, 386, 326, 226, 118, 10, 11, 141, 277
Offset: 0
The array begins:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
1, 6, 13, 22, 33, 46, 61, 78, 97, 118, ...
2, 13, 28, 49, 74, 105, 140, 181, 226, 277, ...
3, 22, 49, 86, 131, 188, 251, 326, 409, 502, ...
4, 33, 74, 131, 200, 289, 386, 503, 632, 777, ...
5, 46, 105, 188, 289, 418, 559, 730, 919, 1132, ...
6, 61, 140, 251, 386, 559, 748, 979, 1234, 1521, ...
7, 78, 181, 326, 503, 730, 979, 1282, 1617, 1994, ...
...
- D. M. Acketa, J. D. Zunic: On the number of linear partitions of the (m,n)-grid. Inform. Process. Lett., 38 (3) (1991), 163-168. See Table A.1.
- 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).
-
V:=proc(m,n) local t1,i,j; t1:=0; for i from 1 to m do for j from 1 to n do if gcd(i,j)=1 then t1:=t1+(m+1-i)*(n+1-j); fi; od; od; t1; end; T:=(m,n)->(2*m*n+m+n+2*V(m,n));
-
V[m_, n_] := Sum[If[GCD[i, j] == 1, (m-i+1)*(n-j+1), 0], {i, 1, m}, {j, 1, n}]; T[m_, n_] := 2*m*n+m+n+2*V[m, n]; Table[T[m-n, n], {m, 0, 11}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)
Showing 1-3 of 3 results.
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