This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357771 #33 Mar 23 2025 20:51:49
%S A357771 1,6,60,744,10224,148896,2250816,34922880,552386304,8867756544,
%T A357771 144044098560,2362292213760,39049785446400,649843233546240,
%U A357771 10876273137008640,182934715370471424,3090181365862170624,52398620697685524480,891492911924665122816,15213249205591283859456,260315328935885892747264
%N A357771 Number of 2n-step closed paths on quasi-regular rhombic (rhombille) lattice starting from a degree-6 node.
%C A357771 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-6 nodes alternate with degree-3 nodes.
%F A357771 a(n) = Sum_{k=0..n} (binomial(n, k) * Sum_{j=0..n} (binomial(n, j) * Sum_{i=0..j} ((1/(2^j))*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i)))). - _Detlef Meya_, May 15 2024
%e A357771 a(2)=60, because there are 6*6=36 paths that visit one of six adjacent vertices, return to the origin, and again visit an adjacent vertex and return to the origin; plus 6*4=24 paths that pass through one of the six vertices at distance 2, leaving and returning via any of two available paths to that vertex; all resulting in a closed path of length 2n=2*2=4.
%t A357771 a[n_] := Sum[Binomial[n, k]*Sum[Binomial[n, j]*Sum[(1/(2^j))*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,0,17}]] (* _Detlef Meya_, May 15 2024 *)
%o A357771 (PARI) a(n) = sum(k=0, n, binomial(n, k) * sum(j=0, n, binomial(n, j) * sum(i=0, j, (1/(2^j)*binomial(2*i, j)*binomial(2*i, i)*binomial(2*(j-i), j-i))))); \\ _Michel Marcus_, May 20 2024
%Y A357771 The accompanying sequences for the number of paths that return to a degree-3 node is A357770.
%Y A357771 Similar sequences for square, hexagonal, and honeycomb lattices are A002894, A002898 and A002893.
%K A357771 nonn,easy,walk
%O A357771 0,2
%A A357771 _Dave R.M. Langers_, Oct 12 2022
%E A357771 More terms from _Detlef Meya_, May 15 2024