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%I #10 Mar 30 2020 17:47:50
%S 3,3,3,3,3,4,2,3,3,3,3,3,3,3,3,2,3,4,3,3,3,3,3,3,3,2,4,3,3,3,3,3,4,2,
%T 3,3,3,3,3,3,3,4,2,3,4,3,3,3,2,4,3,2,3,4,3,3,3,2,3,4,2,3,3,3,3,3,2,4,
%U 3,3,3,3,3,3,3,3,4,3,2,3,3,3,4,2,3,4,3
%N Sizes of maximal weakly increasing subsequences of A000002.
%F a(n) = A000002(2*n - 1) + A000002(2*n).
%e The weakly increasing subsequences begin: (1,2,2), (1,1,2), (1,2,2), (1,2,2), (1,1,2), (1,1,2,2), (1,2), (1,1,2), (1,2,2), (1,1,2), (1,1,2), (1,2,2), (1,2,2).
%t kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
%t kol[n_Integer]:=Nest[kolagrow,{1},n-1];
%t Length/@Split[kol[40],#1<=#2&]
%Y The number of runs in the first n terms of A000002 is A156253.
%Y The weakly decreasing version is A332273.
%Y Cf. A000002, A001462, A013947, A013948, A088568, A288605, A296658, A329315, A329316, A329317, A329362.
%K nonn
%O 1,1
%A _Gus Wiseman_, Mar 08 2020
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