A355048 Number of unoriented orthoplex n-ominoes with cell centers determining n-3 space.

3, 18, 122, 655, 3240, 14531, 61520, 247381, 958434, 3598594, 13180348, 47274577, 166642096, 578750970, 1984671466, 6731351834, 22612409886, 75321920403, 249028297179, 817867225710, 2670093233760, 8670380548402

Offset: 6

Author

Robert A. Russell, Jun 16 2022

Keywords

• nonn
• easy

Comments

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. For unoriented polyominoes, chiral pairs are counted as one.

Examples

			a(6)=3 because there are 3 hexominoes in 2^3 space. The two vacant cells share just a face, an edge, or a vertex.
		

Links

• Robert A. Russell, Table of n, a(n) for n = 6..100
• Robert A. Russell, Trunk Generating Functions

Crossrefs

Cf. A355047 (oriented), A355049 (chiral), A355050 (achiral) A355051 (asymmetric), A000081 (rooted trees).

Other dimensions: A036367 (n-2), A000055 (n-1), A355053 (multidimensional).

Programs

Mathematica

sb[n_,k_] := sb[n,k] = b[n+1-k,1] + If[n<2k,0,sb[n-k,k]];
b[1,1] := 1; b[n_,1] := b[n,1] = Sum[b[i,1]sb[n-1,i]i,{i,1,n-1}]/(n-1);
b[n_,k_] := b[n,k] = Sum[b[i,1]b[n-i,k-1],{i,1,n-1}];
nmax = 30; B[x_] := Sum[b[i,1]x^i,{i,0,nmax}]
Drop[CoefficientList[Series[(14B[x]^6 + 3B[x]^7 + 6B[x]^4B[x^2] + 6B[x]^5B[x^2] + 18B[x]^2B[x^2]^2 + 3B[x]^3B[x^2]^2 + 26B[x^2]^3 + 6 B[x]B[x^2](B[x^2]^2 + B[x^4]) + 4B[x^3]^2 + 4B[x^6]) / 24 + B[x]^3 (38B[x]^4 + 9B[x]^5 + 4B[x]^2B[x^2] + 10B[x]^3B[x^2] + 2B[x^2]^2 + B[x]B[x^2]^2) / (8(1-B[x])) + B[x]^6 (16B[x]^2 + 6B[x]^3 + B[x^2] + B[x] (5 + 2B[x^2])) / (2(1-B[x])^2) + B[x]^7 (2 + 42B[x] + 51B[x]^2 + 24B[x]^3 + 3B[x^2]) / (12(1-B[x])^3) + B[x]^9 (17 + 8B[x]) / (8(1-B[x])^4) + 3B[x]^10 / (8(1-B[x])^5) + B[x^2]^2(B[x]^4 + 4B[x]^2 B[x^2] + 12B[x^2]^2 + B[x^4] + B[x] (8B[x^2] + 5B[x^2]^2 + B[x^4])) / (4(1-B[x^2])) + B[x^2]^4 (8 + 16B[x^2] + B[x] (19 + 8B[x^2])) / (8(1-B[x^2])^2) + 3(1 + B[x])B[x^2]^5 / (4(1-B[x^2])^3) + 2B[x]B[x^3]^2 / (6(1-B[x^3])) + B[x]B[x^4]^2 / (4(1-B[x^4])) + B[x]^2B[x^2]^2(5B[x]^3 + 2B[x^2] + B[x](2 + B[x^2])) / (4(1-B[x])(1-B[x^2])) + B[x]^5(1+4B[x])B[x^2]^2 / (4(1-B[x])^2(1-B[x^2])) + B[x]^6 B[x^2]^2 / (4(1-B[x])^3(1-B[x^2])) + 3B[x]^2B[x^2]^4 / (8(1-B[x])(1-B[x^2])^2) + B[x^2](1+B[x])B[x^4]^2 / (4(1-B[x^2])(1-B[x^4])), {x,0,nmax}],x],6]

Formula

a(n) = A355047(n) - A355049(n) = (A355047(n) + A355050(n)) / 2 = A355049(n) + A355050(n).

G.f.: (14*B(x)^6 + 3*B(x)^7 + 6*B(x)^4*B(x^2) + 6*B(x)^5*B(x^2) + 18*B(x)^2*B(x^2)^2 + 3*B(x)^3*B(x^2)^2 + 26*B(x^2)^3 + 6*B(x)*B(x^2)*(B(x^2)^2 + B(x^4)) + 4*B(x^3)^2 + 4*B(x^6)) / 24 + B(x)^3*(38*B(x)^4 + 9*B(x)^5 + 4*B(x)^2*B(x^2) + 10*B(x)^3*B(x^2) + 2*B(x^2)^2 + B(x)*B(x^2)^2) / (8*(1-B(x))) + B(x)^6*(16*B(x)^2 + 6*B(x)^3 + B(x^2) + B(x)*(5 + 2*B(x^2))) / (2*(1-B(x))^2) + B(x)^7*(2 + 42*B(x) + 51*B(x)^2 + 24*B(x)^3 + 3*B(x^2)) / (12*(1-B(x))^3) + B(x)^9*(17 + 8*B(x)) / (8*(1-B(x))^4) + 3*B(x)^10 / (8*(1-B(x))^5) + B(x^2)^2*(B(x)^4 + 4*B(x)^2*B(x^2) + 12*B(x^2)^2 + B(x^4) + B(x)*(8*B(x^2) + 5*B(x^2)^2 + B(x^4))) / (4*(1-B(x^2))) + B(x^2)^4*(8 + 16*B(x^2) + B(x)*(19 + 8*B(x^2))) / (8*(1-B(x^2))^2) + 3*(1 + B(x))*B(x^2)^5 / (4*(1-B(x^2))^3) + 2*B(x)*B(x^3)^2 / (6*(1-B(x^3))) + B(x)*B(x^4)^2 / (4*(1-B(x^4))) + B(x)^2*B(x^2)^2*(5*B(x)^3 + 2*B(x^2) + B(x)*(2 + B(x^2))) / (4*(1-B(x))*(1-B(x^2))) + B(x)^5*(1+4*B(x))*B(x^2)^2 / (4*(1-B(x))^2*(1-B(x^2))) + B(x)^6*B(x^2)^2 / (4*(1-B(x))^3*(1-B(x^2))) + 3*B(x)^2*B(x^2)^4 / (8*(1-B(x))*(1-B(x^2))^2) + B(x^2)*(1+B(x))*B(x^4)^2 / (4*(1-B(x^2))*(1-B(x^4))), where B(x) is the generating function for rooted trees with n nodes in A000081.