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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. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
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.
LINKS
EXAMPLE
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)
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;
PROG
(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
CROSSREFS
Sequence in context: A342514 A238571 A325874 * A238687 A238433 A238424
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Nov 03 2017
EXTENSIONS
a(19)-a(36) from Don Reble, Nov 04 2017
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)