A338958
Number of chiral pairs of colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.
Original entry on oeis.org
68774446614978208476646592, 5523164445430504871588714239322107782006441, 5448873034167734394145221152621861950913444709790439644, 10956401434158576570935650756489255491646473924447332613392130825
Offset: 2
Cf.
A338956 (oriented),
A338957 (unoriented),
A338959 (achiral),
A338954 (up to n colors),
A338950 (vertices, facets),
A331352 (5-cell),
A331360 (8-cell edges, 16-cell faces),
A331356 (16-cell edges, 8-cell faces),
A338982 (120-cell, 600-cell).
-
bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (*binomial series*)
Drop[CoefficientList[bp[8]/12+bp[12]/8-bp[16]/24-bp[18]/18-bp[20]/6-5bp[24]/96+bp[32]/24+bp[36]/36-5bp[48]/1152+bp[50]/16-bp[52]/96-bp[60]/96+bp[96]/1152,x],2]
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:
Comments