cp's OEIS Frontend

A357770 Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-3 node.

%I A357770 #25 Mar 23 2025 20:51:46
%S A357770 1,3,30,372,5112,74448,1125408,17461440,276193152,4433878272,
%T A357770 72022049280,1181146106880,19524892723200,324921616773120,
%U A357770 5438136568504320,91467357685235712,1545090682931085312,26199310348842762240,445746455962332561408,7606624602795641929728
%N A357770 Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-3 node.
%C A357770 Paths that return to the same point in a quasi-regular rhombic lattice must always have even length (i.e., 2n) because of parity: degree-3 nodes alternate with degree-6 nodes.
%F A357770 a(0) = 1; a(n) = Sum_{k=0..n} (binomial(n, k) * Sum_{j=0..n} (binomial(n, j) * Sum_{i=0..j} ((1/(2^(j + 1)))*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i)))). - _Detlef Meya_, May 20 2024
%e A357770 a(2)=30, because there are 3*3=9 paths that visit one of three adjacent vertices, return to the origin, and again visit an adjacent vertex and return to the origin; 3*5=15 paths visiting one of five distance-2 vertices that are adjacent to the three adjacent vertices; plus 3*2=6 paths traversing the perimeter of three adjacent rhombi in counterclockwise or clockwise direction; all resulting in a closed path of length 2n=2*2=4.
%t A357770 a[0] := 1; a[n_] := Sum[Binomial[n, k]*Sum[Binomial[n, j]*Sum[(1/(2^(j+1)))*Binomial[2*i, j]*Binomial[2*i, i]*Binomial[2*(j-i), j-i], {i, 0, j}], {j, 0, n}], {k, 0, n}]; Flatten[Table[a[n], {n, 0, 19}]] (* _Detlef Meya_, May 20 2024 *)
%Y A357770 The accompanying sequences for the number of paths that return to a degree-6 node is A357771.
%Y A357770 Similar sequences for square, hexagonal, and honeycomb lattices are A002894, A002898 and A002893.
%K A357770 nonn,easy,walk
%O A357770 0,2
%A A357770 _Dave R.M. Langers_, Oct 12 2022
%E A357770 More terms from _Detlef Meya_, May 20 2024