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, 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% |

+--------------+-------------------------+------------+

Links

Table of n, a(n) for n=1..88.

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

Adjacent sequences: A306343 A306344 A306345 * A306347 A306348 A306349

Keyword

nonn,tabf

Author

Michel Lagneau, Feb 09 2019

Status

approved