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A293957 When A002487 is written as a triangle the n-th row has length 2^(n-1); a(n) is the maximal multiplicity of any entry in that row, considering the entries strictly between the initial 1 and the central 2.

%I A293957 #34 Jun 20 2022 10:12:27
%S A293957 0,0,0,1,1,2,2,4,5,6,8,12,16,22,29,36,48,67,84,118,151,203,270,362,
%T A293957 472,636,846,1142,1526,2024,2736,3666,4918,6550,8776,11796,15824
%N A293957 When A002487 is written as a triangle the n-th row has length 2^(n-1); a(n) is the maximal multiplicity of any entry in that row, considering the entries strictly between the initial 1 and the central 2.
%C A293957 The maximal entry is row n is Fibonacci(n+1), and the smallest missing number is A135510(n). The number of distinct numbers in each row is given by A293160.
%C A293957 It would be nice to have a formula for this sequence, or at least some bounds.
%e A293957 Rows 0 through 6 of A002487 are:
%e A293957 0,
%e A293957 1,
%e A293957 1, 2,
%e A293957 1, 3, 2, 3,
%e A293957 1, 4, 3, 5, 2, 5, 3, 4,
%e A293957 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5,
%e A293957 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6,
%e A293957 To find a(5) we consider the entries 1, 5, 4, 7, 3, 8, 5, 7, 2 in row 5. Ignoring the initial 1 and the final 2, the maximal multiplicity is 2 (for example, 5 appears twice), so a(5) = 2.
%e A293957 From _Don Reble_, Nov 04 2017: (Start)
%e A293957 The initial values of a(n) for n >= 3 together with the terms that have the highest multiplicity are:
%e A293957 3    1 [3]
%e A293957 4    1 [3 4 5]
%e A293957 5    2 [5 7]
%e A293957 6    2 [5 7 9 11]
%e A293957 7    4 [11]
%e A293957 8    5 [13 17]
%e A293957 9    6 [19 23 31 41]
%e A293957 10    8 [23 37 43]
%e A293957 11   12 [71]
%e A293957 12   16 [71]
%e A293957 13   22 [127]
%e A293957 14   29 [109]
%e A293957 15   36 [199 251]
%e A293957 16   48 [263]
%e A293957 17   67 [433]
%e A293957 18   84 [701]
%e A293957 19  118 [839]
%e A293957 20  151 [1193]
%e A293957 21  203 [1801]
%e A293957 22  270 [2693]
%e A293957 23  362 [4229]
%e A293957 24  472 [4349]
%e A293957 25  636 [7759]
%e A293957 26  846 [11287]
%e A293957 27 1142 [14627]
%e A293957 28 1526 [20929]
%e A293957 29 2024 [37243]
%e A293957 30 2736 [43133]
%e A293957 31 3666 [67231]
%e A293957 32 4918 [90227]
%e A293957 33 6550 [127819]
%e A293957 34 8776 [181031]
%e A293957 35 11796 [251071]
%e A293957 36 15824 [394549]
%e A293957 (End)
%p A293957 A002487 := proc(n) option remember; if n <= 1 then n elif n mod 2 = 0 then procname(n/2); else procname((n-1)/2)+procname((n+1)/2); fi; end:
%p A293957 ans:=[];
%p A293957 for n from 3 to 18 do
%p A293957 b1:=2^(n-1); b2:=2^n-1; b3:=2^(n-2)-1; mx:=0;
%p A293957 ar:=Array(0..b1-1,0);
%p A293957 for k from 1 to b3 do
%p A293957 kk:=b1+k;
%p A293957 v:=A002487(kk);
%p A293957 ar[v]:=ar[v]+1;
%p A293957 od:
%p A293957    for k from 0 to b1-1 do if ar[k]>mx then mx:=ar[k]; fi; od:
%p A293957 ans:=[op(ans),mx];
%p A293957 od:
%p A293957 ans;
%o A293957 (Python)
%o A293957 from itertools import chain, product
%o A293957 from collections import Counter
%o A293957 from functools import reduce
%o A293957 def A293957(n): return 0 if n <= 2 else max(Counter(m for m in (sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if y else (x[0]+x[1],x[1]),chain(k,(1,)),(1,0))) for k in product((False,True),repeat=n-2)) if m != 1 and m != 2).values()) # _Chai Wah Wu_, Jun 20 2022
%Y A293957 Cf. A002487, A049456, A135510, A293160.
%K A293957 nonn,more
%O A293957 0,6
%A A293957 _N. J. A. Sloane_, Nov 03 2017
%E A293957 a(19)-a(36) from _Don Reble_, Nov 04 2017