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A317444 Number of permutations of [n] whose lengths of increasing runs are distinct Fibonacci numbers.

%I A317444 #13 Jul 14 2021 02:47:30
%S A317444 1,1,1,5,6,19,212,40,757,2170,13546,379084,8978,73195,2702092,772852,
%T A317444 38833826,213557110,2390871412,150689939006,9394670,634504029,
%U A317444 4522073096,63395566566,5160905755362,192831696582,3068824154606,289158899744046,116561588867106
%N A317444 Number of permutations of [n] whose lengths of increasing runs are distinct Fibonacci numbers.
%H A317444 Alois P. Heinz, <a href="/A317444/b317444.txt">Table of n, a(n) for n = 0..150</a>
%p A317444 g:= (n, s)-> `if`(n in s or not
%p A317444     (issqr(5*n^2+4) or issqr(5*n^2-4)), 0, 1):
%p A317444 b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
%p A317444       `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
%p A317444        , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
%p A317444     end:
%p A317444 a:= n-> b(n, 0$2, {}):
%p A317444 seq(a(n), n=0..30);
%t A317444 g[n_, s_] := If[MemberQ[s, n] || !(
%t A317444      IntegerQ@Sqrt[5*n^2 + 4] || IntegerQ@Sqrt[5*n^2 - 4]), 0, 1];
%t A317444 b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
%t A317444      If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
%t A317444      {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
%t A317444 a[n_] := b[n, 0, 0, {}];
%t A317444 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 14 2021, after _Alois P. Heinz_ *)
%Y A317444 Cf. A000045, A317128, A317445, A317446, A317447, A317448.
%K A317444 nonn
%O A317444 0,4
%A A317444 _Alois P. Heinz_, Jul 28 2018