A357484 Number of linearity regions of a max-pooling function with a 3 by n input and 2 by 2 pooling windows.

1, 14, 150, 1536, 15594, 158050, 1601356, 16223814, 164366170, 1665216896, 16870539234, 170917714410, 1731590444316, 17542976546494, 177730263461890, 1800609290091936, 18242215773029194, 184814350419581330, 1872379131238643436, 18969325721395559574

Offset: 1

Author

Alejandro H. Morales, Sep 30 2022

Keywords

• nonn
• easy

Comments

a(n) is also the number of vertices of the Minkowski sum of 2*n-2 simplices conv(e_{i,j},e_{i,j+1},e_{i+1,j},e_{i+1,j+1}) for i=0,1 and j=0,...,n-2, viewing R^(3n) having basis {e_{i,j} | i=0,1,2; j=0,...,n-1}.

Examples

			For n = 2 the a(2)=14 vertices are (00,10), (00,11), (00,20), (00,21), (01,10), (01,11), (01,20), (01,21), (10,10), (10, 20), (10,21), (11,11), (11, 20), (11, 21), where (ij,kl) represents e_{i,j}+e_{k,l}. The pair (10,11) does not represent vertices since e_{1,0}+e_{1,1} is a convex combination of the vectors 2e_{1,0} + 2e_{1,1}. Ditto for the pair (11,10).
		

Links

• Laura Escobar, Patricio Gallardo, Javier González-Anaya, José L. González, Guido Montúfar, and Alejandro H. Morales, Enumeration of max-pooling responses with generalized permutohedra, arXiv:2209.14978 [math.CO], 2022.
• Index entries for linear recurrences with constant coefficients, signature (13,-31,20,-4).

Crossrefs

Cf. A007070, A033303.

Programs

Maple

a:= proc(n) option remember;
   if n = 1 then
     return(1);
   elif n = 2 then
     return(14);
   elif n = 3 then
     return(150);
   elif n = 4 then
     return(1536);
   else
     return(13*a(n-1) - 31*a(n-2) + 20*a(n-3) - 4*a(n-4));
   end if;
end proc:
seq(a(n), n=1..20);

Mathematica

LinearRecurrence[{13, -31, 20, -4}, {1, 14, 150, 1536}, 20] (* Hugo Pfoertner, Oct 05 2022 *)

Sage

@cached_function
def a(n):
    if n < 5: return [1, 14, 150, 1536][n - 1]
    return 13*a(n-1) - 31*a(n-2) + 20*a(n-3) - 4*a(n-4)
print([a(n) for n in range(1, 21)])

Formula

a(n) = 13*a(n-1) - 31*a(n-2) + 20*a(n-3) - 4*a(n-4) for n>= 5.

G.f.: (x+x^2-x^3)/(1-13*x+31*x^2-20*x^3+4*x^4).