# A sequence related to Lionel Levile's sequence?

Paul D. Hanna pauldhanna at juno.com
Fri Nov 4 18:58:58 CET 2005

```Seqfans,
I am on my lunch break, so I do not have time to develop the
following
(would someone care to finish this NEW sequence?).

of generations of nodes of a tree, especially in light of
Alexandre's connection with Golomb's seq. A001462.

If we form a triangle (irregular shape) out of A001462
where each row n is the n-th generation of terms, like so:

1;
2,2;
3,3;
4,4,4,5,5,5;
6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11;
etc.

The triangle lists the generations of a tree-like structure in the
following way.
Row 4: [4,4,4,5,5,5] is generated from row 3: [3,3]
because there are (3) 4's and (3) 5's in row 4.
Likewise row 5 is generated from row 4: [4,4,4,5,5,5]
because there are (4) 6's, (4) 7's, (4) 8's, (5) 9's, (5) 10's, (5) 11's
in row 5.
etc.

Now, the row sums form a new sequence (offset=1):

{1,4,6,27,234,6202, ...}

where there exist 3 different ways of coming up with the same numbers:

(1) number of terms in each row
(one exception: a(2) is to equal the number of terms in rows 2 and 3
together);

(2) row sums:
a(2) = 4 = 2 + 2;
a(3) = 6 = 3 + 3;
a(4) = 27 = 4 + 4 + 4 + 5 + 5 + 5;
a(5) = 234 =
6+6+6+6+7+7+7+7+8+8+8+8+9+9+9+9+9+10+10+10+10+10+11+11+11+11+11.

(3) weighted row sums:
a(2) = 4 = (2)*2;
a(3) = 6 = (2)*3;
a(4) = 27 = (3)*4 + (3)*5;
a(5) = 234 = (4)*6 + (4)*7 + (4)*8 + (5)*9 + (5)*10 + (5)*11.

I feel that there is also a matrix recurrence that can generate this
sequence ...
so I would like to know how the rest of the sequence continues.

Would someone please generate more terms and submit the sequence?

Thanks,
Paul

>---------------------------------------------------------------------
Alexandre Wajnberg wrote:

Looks like Golomb's sequence A001462:

ID Number: A001462 (Formerly M0257 and N0091)
URL:       http://www.research.att.com/projects/OEIS?Anum=A001462
Sequence:  1,2,2,3,3,4,4,4,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,9,10,
10,10,10,10,11,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,
13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,16,
17,17,17,17,17,17,17,18,18,18,18,18,18,18,19
Name:      Golomb's sequence: a(n) is the number of times n occurs,
starting
with a(1) = 1.
Comments:  It is understood that a(n) is taken to be the smallest number
>=
a(n-1) which is compatible with the description.
Also called Silverman's sequence.
Vardi gives several identities satisfied by A001463 and this
sequence.

```