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a(n) = Sum_{k=0..n-1} A356692(n-1,k) for n >= 1. - _Alois P. Heinz_, Aug 28 2022
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b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[Sequence @@ Sort[{o-j, u+j-1}]], {j, 1, Min[2, o]}] + Sum[b[Sequence @@ Sort[{u-j, o+j-1}]], {j, 1, Min[2, u]}]]; a[n_] := If[n == 0, 1, Sum[b[Sequence @@ Sort[{j-1, n-j}]], {j, 1, n}]]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)
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a(n) ~ c * 4^n, where c = 0.052940679853652794231561081876002147090052503777... - Vaclav Kotesovec, Feb 23 2014
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