A332595
Number of triangular regions in a "frame" of size n X n (see Comments in A331776 for definition), divided by 4.
Original entry on oeis.org
1, 12, 32, 72, 128, 232, 368, 576, 832, 1232, 1712, 2328, 3040, 4040, 5184, 6616, 8224, 10248, 12496, 15144, 18048, 21688, 25664, 30184, 35072, 41000, 47392, 54608, 62336, 71088, 80416, 90864, 101952, 114832, 128480, 143352, 159040, 176984, 195888, 216424, 237984, 261624, 286384, 313184, 341184, 372496, 405184
Offset: 1
-
V := proc(m,n,q) local a,i,j; a:=0;
for i from 1 to m do for j from 1 to n do
if gcd(i,j)=q then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
f := n -> if n=1 then 4 else 8*n^2 + 16*n - 24 + 8*V(n,n,2); fi;
[seq(f(n)/4, n=1..60)]; # N. J. A. Sloane, Mar 09 2020
A332610
Triangle read by rows: T(m,n) = number of triangular regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).
Original entry on oeis.org
4, 14, 48, 32, 102, 128, 70, 192, 204, 288, 124, 326, 312, 396, 512, 226, 524, 516, 600, 716, 928, 360, 802, 784, 868, 984, 1196, 1472, 566, 1192, 1196, 1280, 1396, 1608, 1884, 2304, 820, 1634, 1704, 1788, 1904, 2116, 2392, 2812, 3328, 1218, 2296, 2500, 2584, 2700, 2912, 3188, 3608, 4124, 4928
Offset: 1
Triangle begins:
[4],
[14, 48],
[32, 102, 128],
[70, 192, 204, 288],
[124, 326, 312, 396, 512],
[226, 524, 516, 600, 716, 928],
[360, 802, 784, 868, 984, 1196, 1472],
[566, 1192, 1196, 1280, 1396, 1608, 1884, 2304],
[820, 1634, 1704, 1788, 1904, 2116, 2392, 2812, 3328],
[1218, 2296, 2500, 2584, 2700, 2912, 3188, 3608, 4124, 4928],
[1696, 3074, 3456, 3540, 3656, 3868, 4144, 4564, 5080, 5884, 6848],
[2310, 4052, 4684, 4768, 4884, 5096, 5372, 5792, 6308, 7112, 8076, 9312],
...
A332611
Triangle read by rows: T(m,n) = number of quadrilateral regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).
Original entry on oeis.org
0, 2, 8, 14, 36, 80, 34, 92, 144, 208, 90, 194, 280, 356, 504, 154, 336, 432, 520, 680, 856, 288, 554, 724, 824, 996, 1184, 1512, 462, 812, 1096, 1208, 1392, 1592, 1932, 2352, 742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640, 1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016
Offset: 1
Triangle begins:
[0],
[2, 8],
[14, 36, 80],
[34, 92, 144, 208],
[90, 194, 280, 356, 504],
[154, 336, 432, 520, 680, 856],
[288, 554, 724, 824, 996, 1184, 1512],
[462, 812, 1096, 1208, 1392, 1592, 1932, 2352],
[742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640],
[1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016],
[1512, 2508, 3268, 3416, 3636, 3872, 4248, 4704, 5372, 6072, 7128],
[2074, 3252, 4416, 4576, 4808, 5056, 5444, 5912, 6592, 7304, 8372, 9616],
....
Showing 1-3 of 3 results.
Comments