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.

0, 0, 0, 1, 1, 2, 2, 4, 5, 6, 8, 12, 16, 22, 29, 36, 48, 67, 84, 118, 151, 203, 270, 362, 472, 636, 846, 1142, 1526, 2024, 2736, 3666, 4918, 6550, 8776, 11796, 15824

Offset: 0

Author

N. J. A. Sloane, Nov 03 2017

Keywords

• nonn
• more

Comments

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.

It would be nice to have a formula for this sequence, or at least some bounds.

Examples

			Rows 0 through 6 of A002487 are:
0,
1,
1, 2,
1, 3, 2, 3,
1, 4, 3, 5, 2, 5, 3, 4,
1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5,
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,
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.
From _Don Reble_, Nov 04 2017: (Start)
The initial values of a(n) for n >= 3 together with the terms that have the highest multiplicity are:
3    1 [3]
4    1 [3 4 5]
5    2 [5 7]
6    2 [5 7 9 11]
7    4 [11]
8    5 [13 17]
9    6 [19 23 31 41]
10    8 [23 37 43]
11   12 [71]
12   16 [71]
13   22 [127]
14   29 [109]
15   36 [199 251]
16   48 [263]
17   67 [433]
18   84 [701]
19  118 [839]
20  151 [1193]
21  203 [1801]
22  270 [2693]
23  362 [4229]
24  472 [4349]
25  636 [7759]
26  846 [11287]
27 1142 [14627]
28 1526 [20929]
29 2024 [37243]
30 2736 [43133]
31 3666 [67231]
32 4918 [90227]
33 6550 [127819]
34 8776 [181031]
35 11796 [251071]
36 15824 [394549]
(End)
		

Crossrefs

Cf. A002487, A049456, A135510, A293160.

Programs

Maple

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:
ans:=[];
for n from 3 to 18 do
b1:=2^(n-1); b2:=2^n-1; b3:=2^(n-2)-1; mx:=0;
ar:=Array(0..b1-1,0);
for k from 1 to b3 do
kk:=b1+k;
v:=A002487(kk);
ar[v]:=ar[v]+1;
od:
   for k from 0 to b1-1 do if ar[k]>mx then mx:=ar[k]; fi; od:
ans:=[op(ans),mx];
od:
ans;

Python

from itertools import chain, product
from collections import Counter
from functools import reduce
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

Extensions

a(19)-a(36) from Don Reble, Nov 04 2017