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A306346
Irregular array read by rows: row 1 = [0]; for n > 1, if row n = [r_1,...,r_k] then row n+1 contains [row n, r_(k+1),..., r_q] where each r_i for i = k+1,...,q is the number of consecutive identical integers counted from left to right in the row n.
0
0, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 7, 1, 1, 1, 1, 1, 3, 1, 7, 1
OFFSET
1,5
COMMENTS
Conjecture 1: a(n) <= 7.
Conjecture 2: a(n) is odd for n > 1 and when a(n) = 3, 5 or 7, n is odd.
The algorithm for generating a(n) consists of the following steps:
Start with row 1 = [0];
row 1 has only one 0, so row 2 = [row 1, 1] = [0, 1];
row 2 has one 1 and one 0, so row 3 = [row 2, 1, 1] = [0, 1, 1, 1];
row 3 has three consecutive 1 and one 0, so row 4 = [Row 3, 3, 1] = [0, 1, 1, 1, 3, 1];
...
Property of the sequence:
a(n) = 3 for the odd values n = 5, 9, 13, 15, 21, 23, 25, 37, 39, ...
a(n) = 5 for the odd values n = 33, 57, 75, 93, 111, 115, 129, 147, ...
a(n) = 7 for the odd values n = 51, 79, 87, 121, 133, 141, 185, 203, ...
Remark:
If row 1 = [1], the sequence a(n) becomes b(n) = 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, ... and it is conjectured that b(n) <= 7.
Statistics for n <= 10^5:
+--------------+-------------------------+------------+
| a(n) | number of occurrences | percentage |
| | for n <= 10^5 | |
+--------------+-------------------------+------------+
| 1 | 72244 | 72.244% |
| 3 | 18030 | 18.030% |
| 5 | 6806 | 6.806% |
| 7 | 2919 | 2.919% |
+--------------+-------------------------+------------+
EXAMPLE
row 5 = [ 0, 1, 1, 1, 3, 1, 1, 1, 3, 1 ]
\_/ \______/ \_/ \______/ \_/ \_/
1 3 1 3 1 1
row 6 = [ row 5, 1, 3, 1, 3, 1, 1 ]
Irregular array starts:
[0];
[0, 1];
[0, 1, 1, 1];
[0, 1, 1, 1, 3, 1];
[0, 1, 1, 1, 3, 1, 1, 1, 3, 1];
[0, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1];
[0, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1];
...
MAPLE
n0:=1:T:=array(1..1000, [0$1000]):
for n from 1 to 50 do :
it:=1:i:=n0:
for k from n0 by -1 to 2 do:
if T[k]=T[k-1]
then
it:=it+1:
else
i:=i+1:T[i]:=it:it:=1:
fi:
od:
i:=i+1:T[i]:=1:n0:=i:
od:
print(T):
CROSSREFS
Cf. A000002.
Sequence in context: A368336 A365633 A091842 * A060901 A351545 A087612
KEYWORD
nonn,tabf
AUTHOR
Michel Lagneau, Feb 09 2019
STATUS
approved