Original entry on oeis.org
3, 10, 30, 82, 180, 374, 656, 1126, 1784, 2726, 3916, 5582, 7628, 10354, 13592, 17542, 22212, 28042, 34672, 42762, 51964, 62614, 74496, 88510, 104044, 121914, 141536, 163666, 187944, 215638, 245492, 279262, 316024, 356458, 399900, 447614, 498700, 555354, 615776, 681286
Offset: 1
A332357
Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1*x_1 + a_2*x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of cells (both 3-sided and 4-sided) in the partition, for m >= n >= 1.
Original entry on oeis.org
1, 2, 5, 3, 9, 17, 4, 14, 28, 47, 5, 20, 41, 70, 105, 6, 27, 57, 99, 150, 215, 7, 35, 75, 131, 199, 286, 381, 8, 44, 96, 169, 258, 372, 497, 649, 9, 54, 119, 211, 323, 467, 625, 817, 1029, 10, 65, 145, 258, 396, 574, 769, 1006, 1268, 1563, 11, 77, 173, 309, 475, 689, 923, 1208, 1523, 1878, 2257
Offset: 1
Triangle begins:
1,
2, 5,
3, 9, 17,
4, 14, 28, 47,
5, 20, 41, 70, 105,
6, 27, 57, 99, 150, 215,
7, 35, 75, 131, 199, 286, 381,
8, 44, 96, 169, 258, 372, 497, 649,
9, 54, 119, 211, 323, 467, 625, 817, 1029,
10, 65, 145, 258, 396, 574, 769, 1006, 1268, 1563,
...
- 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. See Theorem 13.
- N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]
A332361
Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1*x_1 + a_2*x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of vertices in the partition, for m >= n >= 1.
Original entry on oeis.org
3, 4, 6, 5, 9, 14, 6, 13, 22, 36, 7, 18, 31, 52, 76, 8, 24, 43, 74, 110, 160, 9, 31, 56, 97, 144, 210, 276, 10, 39, 72, 126, 188, 275, 363, 478, 11, 48, 89, 157, 235, 345, 456, 601, 756, 12, 58, 109, 193, 290, 427, 565, 745, 938, 1164, 13, 69, 130, 231, 347, 511, 675, 890, 1120, 1390, 1660
Offset: 1
Triangle begins:
3,
4, 6,
5, 9, 14,
6, 13, 22, 36,
7, 18, 31, 52, 76,
8, 24, 43, 74, 110, 160,
9, 31, 56, 97, 144, 210, 276,
10, 39, 72, 126, 188, 275, 363, 478,
11, 48, 89, 157, 235, 345, 456, 601, 756,
12, 58, 109, 193, 290, 427, 565, 745, 938, 1164,
...
- 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. See Theorem 13.
- N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]
-
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;
ct3 := proc(m,n) local i; global VR;
if m=1 or n=1 then max(m,n) else VR(m,n,2)/2+m+n+1; fi; end; # A332354
ct4 := proc(m,n) local i; global VR;
if m=1 or n=1 then 0 else VR(m,n,1)/4-VR(m,n,2)/2-m/2-n/2-1; fi; end; # A332356
ct := (m,n) -> ct3(m,n) + ct4(m,n); # A332357
cte := proc(m,n) local i; global VR;
if m=1 or n=1 then 2*max(m,n)+1 else VR(m,n,1)/2-VR(m,n,2)/4+m+n; fi; end; # A332359
ctv := (m,n) -> cte(m,n) - ct(m,n) + 1; # A332361
for m from 1 to 12 do lprint([seq(ctv(m,n),n=1..m)]); od:
Showing 1-3 of 3 results.