A133330 implicit in "On sums of figurate numbers by using techniques of poset representation theory"

Tue Jun 17 06:50:46 CEST 2008

It seems that the following paper of June 16, 2008, implicitly refers
to A133330 Sums of exactly three positive octahedral numbers A005900,
in Corollary 17, p.17. It refers to some other seqs which may or may
not be in OEIS, namely:
(a) natural numbers which are the sum of three polygonal numbers of
positive rank;
(b) natural numbers which are the sum of four cubes with two of them equal.

Are either of those in OEIS?

 http://arxiv.org/pdf/0806.2486
    Title: On sums of figurate numbers by using techniques of poset
representation theory
    Authors: Agustin Moreno C
    Comments: 18 pages, submitted to International Journal of Number Theory
    Subjects: Number Theory (math.NT); Representation Theory (math.RT)

    We use representations and differentiation algorithms of posets,
in order to obtain results concerning unsolved problems on figurate
numbers. In particular, we present criteria for natural numbers which
are the sum of three octahedral numbers, three polygonal numbers of
positive rank or four cubes with two of them equal. Some identities of
the Rogers-Ramanujan type involving this class of numbers are also
obtained.

There would be at least 1, and perhaps 3 comments to submit.  Thought
I'd ask before form submitting. Any suggestions?

Best,

Jonathan Vos Post

jvp> From seqfan-owner at ext.jussieu.fr  Tue Jun 17 06:51:54 2008
jvp> Date: Mon, 16 Jun 2008 21:50:46 -0700
jvp> From: "Jonathan Post" <jvospost3 at gmail.com>
jvp> To: "Sequence Fans" <seqfan at ext.jussieu.fr>
jvp> Subject: A133330 implicit in "On sums of figurate numbers by using techniques of poset representation theory"
jvp> 
jvp> It seems that the following paper of June 16, 2008, implicitly refers
jvp> to A133330 Sums of exactly three positive octahedral numbers A005900,
jvp> in Corollary 17, p.17. It refers to some other seqs which may or may
jvp> not be in OEIS, namely:
jvp> (a) natural numbers which are the sum of three polygonal numbers of
jvp> positive rank;

These are all natural numbers >2, because the 1st and 2nd column (rank 1 and 2)
of A057145 cover that case. The preprint actually asks this for the case of three
polygonal numbers of the same positive rank. For k=3 the numbers are
9=3+3+3
12=3+3+6
15=3+3+9
18=3+3+12
21=3+3+15
24=3+3+18
27=3+3+21
30=3+3+24
33=3+3+27
36=3+3+30
39=3+3+33
42=3+3+36
45=3+3+39
48=3+3+42
51=3+3+45
54=3+3+48
57=3+3+51
60=3+3+54
63=3+3+57
66=3+3+60
69=3+3+63
72=3+3+66
75=3+3+69
78=3+3+72
81=3+3+75
84=3+3+78
87=3+3+81
90=3+3+84
(boring, because P(n,3) are essentially the multiples of 3)
....

For k=4 the numbers are
12=4+4+4
18=4+4+10
24=4+4+16
30=4+4+22
36=4+4+28
42=4+4+34
48=4+4+40
54=4+4+46
60=4+4+52
66=4+4+58
72=4+4+64
78=4+4+70
84=4+4+76
90=4+4+82
96=4+4+88
102=4+4+94
(boring, kind of multiples of 6)

One could ask for numbers which are the sum of three *distinct* polygonal
numbers of the same positive rank. For k=3 we have
18=3+6+9
21=3+6+12
24=3+6+15
27=3+6+18
30=3+6+21
33=3+6+24
36=3+6+27
39=3+6+30
42=3+6+33
45=3+6+36
48=3+6+39
51=3+6+42
54=3+6+45
(boring, see above)

jvp> (b) natural numbers which are the sum of four cubes with two of them equal.

If this is "natural numbers which are the sum of four positive cubes with exactly (no more than)
two of them equal", the sequence is probably not in the OEIS. In this list are
(to be checked by someone with a CAS or pen and paper)

9=2*0^3+1^3+2^3
10=0^3+2*1^3+2^3
17=0^3+1^3+2*2^3
28=2*0^2+1^3+3^3
29=0^3+2*1^3+3^3
35=2*0^2+2^3+3^3
37=2*1^2+2^3+3^3
43=0^3+2*2^3+3^3
44=1^3+2*2^3+3^3
55=0^3+1^3+2*3^3
62=0^3+2^3+2*3^3
63=1^3+2^3+2*3^3
65=2*0^2+1^3+4^3
66=0^3+2*1^3+4^3
72=2*0^2+2^3+4^3
74=2*1^2+2^3+4^3
80=0^3+2*2^3+4^3
81=1^3+2*2^3+4^3
91=2*0^2+3^3+4^3
93=2*1^2+3^3+4^3
107=2*2^2+3^3+4^3
118=0^3+2*3^3+4^3
119=1^3+2*3^3+4^3
126=2*0^2+1^3+5^3
127=0^3+2*1^3+5^3
129=0^3+1^3+2*4^3

If this is "natural numbers which are the sum of four strictly positive cubes with exactly (no more than)
two of them equal", the sequence is probably not in the OEIS. This is a sub-set of above:
37=2*1^2+2^3+3^3
44=1^3+2*2^3+3^3
63=1^3+2^3+2*3^3
74=2*1^2+2^3+4^3
81=1^3+2*2^3+4^3
93=2*1^2+3^3+4^3
107=2*2^2+3^3+4^3
119=1^3+2*3^3+4^3
126=2^3+2*3^3+4^3
I am using "strictly positive" in the sense of A000027, "positive"
in the sense of A001477.

If, alternatively, this is "natural numbers which are the sum of four positive cubes with
at least two of them equal" we get a list very similar (but not equal) to A004826
(note that 36 is missing, for example):
0=2*0^3+0^3+0^3
1=2*0^3+0^3+1^3
2=0^3+0^3+2*1^3
3=0^3+2*1^3+1^3
4=2*1^3+1^3+1^3
8=2*0^3+0^3+2^3
9=2*0^3+1^3+2^3
10=0^3+2*1^3+2^3
11=2*1^3+1^3+2^3
16=0^3+0^3+2*2^3
17=0^3+1^3+2*2^3
18=1^3+1^3+2*2^3
24=0^3+2*2^3+2^3
25=1^3+2*2^3+2^3
27=2*0^3+0^3+3^3
28=2*0^3+1^3+3^3
29=0^3+2*1^3+3^3
30=2*1^3+1^3+3^3
32=2*2^3+2^3+2^3
35=2*0^3+2^3+3^3
37=2*1^3+2^3+3^3
43=0^3+2*2^3+3^3
44=1^3+2*2^3+3^3
51=2*2^3+2^3+3^3
54=0^3+0^3+2*3^3
55=0^3+1^3+2*3^3
56=1^3+1^3+2*3^3
62=0^3+2^3+2*3^3
63=1^3+2^3+2*3^3
64=2*0^3+0^3+4^3
65=2*0^3+1^3+4^3
66=0^3+2*1^3+4^3
67=2*1^3+1^3+4^3
70=2^3+2^3+2*3^3
72=2*0^3+2^3+4^3
74=2*1^3+2^3+4^3
80=0^3+2*2^3+4^3
81=0^3+2*3^3+3^3
82=1^3+2*3^3+3^3
88=2*2^3+2^3+4^3
89=2^3+2*3^3+3^3
91=2*0^3+3^3+4^3
93=2*1^3+3^3+4^3
107=2*2^3+3^3+4^3
108=2*3^3+3^3+3^3
118=0^3+2*3^3+4^3
119=1^3+2*3^3+4^3
125=2*0^3+0^3+5^3
126=2*0^3+1^3+5^3
127=0^3+2*1^3+5^3
128=0^3+0^3+2*4^3
129=0^3+1^3+2*4^3
130=1^3+1^3+2*4^3

If this is "natural numbers which are the sum of four strictly positive cubes with
at least two of them equal", we get "almost" A003327 or A025403 or A047703 (note that
100 seems to be missing):
4=2*1^3+1^3+1^3
11=2*1^3+1^3+2^3
18=1^3+1^3+2*2^3
25=1^3+2*2^3+2^3
30=2*1^3+1^3+3^3
32=2*2^3+2^3+2^3
37=2*1^3+2^3+3^3
44=1^3+2*2^3+3^3
51=2*2^3+2^3+3^3
56=1^3+1^3+2*3^3
63=1^3+2^3+2*3^3
67=2*1^3+1^3+4^3
70=2^3+2^3+2*3^3
74=2*1^3+2^3+4^3
81=1^3+2*2^3+4^3
82=1^3+2*3^3+3^3
88=2*2^3+2^3+4^3
89=2^3+2*3^3+3^3
93=2*1^3+3^3+4^3
107=2*2^3+3^3+4^3
108=2*3^3+3^3+3^3
119=1^3+2*3^3+4^3
126=2^3+2*3^3+4^3
128=2*1^3+1^3+5^3

This business becomes much more demanding if negative cubes are incorporated, and this
is actually what the preprint is referring to.

jvp> 
jvp> Are either of those in OEIS?
jvp> 
jvp>  http://arxiv.org/pdf/0806.2486
jvp>     Title: On sums of figurate numbers by using techniques of poset
jvp> representation theory
jvp>     Authors: Agustin Moreno C
jvp>     Comments: 18 pages, submitted to International Journal of Number Theory
jvp>     Subjects: Number Theory (math.NT); Representation Theory (math.RT)
jvp> ....