# [seqfan] more on bases

David Newman davidsnewman at gmail.com
Mon Mar 23 02:58:21 CET 2009

```Using the terms of A008932, call a set, A,  a basis of order h, if every
number can be written as the sum of h, (not necesarily distinct) elements of
A.  Call a basis an increasing basis of order h if its elements are arranged
in increasing order, a0<a1<a2<...

For example   0, 1, 2, 4, 8, 9, 16,18, 32, 36,... is an increasing basis of
order 3.( This sequence does not seem to be in the OEIS)

This sequence is made as follows: Take the union of the following three
sets: (1)  the set of all nonnegative numbers which can be written in base
two as sums of powers, k, of 2, where k is congruent to 0 mod 3; (2) the set
of all nonnegative numbers which can be written in base two as sums of
powers, k, of 2, where k is congruent to 1 mod 3; (3) the set of all
nonnegative numbers which can be written in base tow as sums of powers, k,
of 2, where k is congruent to 2 mod 3.

Consider the set of all initial subsequences of any length {a0, a1, a2,,,
an} of all the increasing bases.  These can be ordered in the library
ordering, giving, for h=3

0
0,1
0,1,2
0,1,3
0,1,4

How many increasing initial sequences of bases of order 3 are there?

The numbers that I've gotten by hand calculation are  1, 1, 3, 13, 86

I invite anyone who's interested to validate and extend this sequence.

Next would be the same problem for bases of order 4,5, etc.

Another variation is the same as above but with the provision that the sums
must be of distinct elements.  These were called restricted bases, I think,
by Erdos.

```