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A317128 Number of permutations of [n] whose lengths of increasing runs are Fibonacci numbers.

%I A317128 #16 Mar 29 2021 08:00:25
%S A317128 1,1,2,6,23,112,652,4425,34358,299971,2910304,31059715,361603228,
%T A317128 4560742758,61947243329,901511878198,13994262184718,230811430415207,
%U A317128 4030772161073249,74301962970014978,1441745847111969415,29374226224980834077,626971133730275593916
%N A317128 Number of permutations of [n] whose lengths of increasing runs are Fibonacci numbers.
%H A317128 Alois P. Heinz, <a href="/A317128/b317128.txt">Table of n, a(n) for n = 0..400</a>
%p A317128 g:= n-> (t-> `if`(issqr(t+4) or issqr(t-4), 1, 0))(5*n^2):
%p A317128 b:= proc(u, o, t) option remember; `if`(u+o=0, g(t),
%p A317128       `if`(g(t)=1, add(b(u-j, o+j-1, 1), j=1..u), 0)+
%p A317128        add(b(u+j-1, o-j, t+1), j=1..o))
%p A317128     end:
%p A317128 a:= n-> b(n, 0$2):
%p A317128 seq(a(n), n=0..27);
%t A317128 g[n_] := With[{t = 5n^2}, If[IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4], 1, 0]];
%t A317128 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, g[t],
%t A317128      If[g[t] == 1, Sum[b[u - j, o + j - 1, 1], {j, 1, u}], 0] +
%t A317128      Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]];
%t A317128 a[n_] := b[n, 0, 0];
%t A317128 a /@ Range[0, 27] (* _Jean-François Alcover_, Mar 29 2021, after _Alois P. Heinz_ *)
%Y A317128 Cf. A000045, A097597, A273001, A317111, A317129, A317130, A317131, A317132, A317444.
%K A317128 nonn
%O A317128 0,3
%A A317128 _Alois P. Heinz_, Jul 21 2018