Conjecture: 29 is largest integer <> h-gonal(i) + j-gonal(k), h, i, j, k >2

[Jonathan Post]

Consider nontrivial sums of exactly two polygonal numbers, as (for
instance) in the table A086271. Nontrivial as follows.

k-gonal(0) = 0 for all k; we don't want these in the sum or we
ambiguate "the number of ways an integer can be represented as the sum
of exactly two polygonal numbers."

k-gonal(1) = 1 for all k; we don't want these in the sum, or trivially
every integer N = (N-1)-gonal(2) + k-gonal(1) for all k.

k-gonal(2) = k for all k, as mentioned above. It is not very
interesting to me to partition integers into two integers => 3.

Hence we consider the set {k-gonal(n) for k, n > 2}

n\k.|....3.....4.....5.....6.....7......8......9....10....11....12....13....14...
3....|....6.....9...12...15...18....21....24....27....30....33....36....39
4....|..10...16...22...28...34....40....46....52....58....64....70....76
5....|..15...25...35...45...55....65....75....85....95..105..115..125
6....|..21...36...51...66...81....96..111..126..141..156..171..186
7....|..28...49...70...91.112..133..154..175..196..217..238..259
8....|..36...64...92.120.148..176..204..232..260..288..316..344
9....|..45...81.117.153.189..225..261..297..333..369..405..441
10..|..55.100.145.190.235..280..325..370..415..460..505..550

We can make all possible sums of pairs of these, and sort them, seeing
that some can be done in more than one way:
12 = 6 + 6
15 = 9 + 6
16 = 10 + 6
18 = 12 + 6 = 9 + 9
19 = 10 + 9
20 = 10 + 10
21 = 15 + 6 = 12 + 9
22 = 16 + 6 = 12 + 10
24 = 18 + 6 = 12 + 12
25 = 16 + 9 = 15 + 10
26 = 16 + 10
27 = 21 + 6 = 18 + 9 = 15 + 12
28 = 22 + 6 = 18 + 10
30 = 24 + 6 = 21 + 9 = 15 + 15
31 = 25 + 6 = 16 + 15
32 = 22 + 10 = 16 + 16
33 = 27 + 6 = 24 + 9 = 18 + 15
34 = 28 + 6 = 25 + 9 = 24 + 10 = 18 + 16
and so forth.

We find the complement to the sum set:

Integers which cannot be written as the sum of two polygonal numbers
of indices > 2:
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 23, 29, ...|

In a few minute's inspection, it seems to me that the number of ways
of representing an integer in our sum set grows (irregularly, to be
sure) fast enough that 29 may be the last such non-represented number.
 If not, I still reasonably conjecture that almost all positive
integers can be so represented.

Any thoughts on this, which seems not to be in OEIS according to
several searches I've tried?  Again, apologies for any errors made in
my having done the above "by hand."

Merry Christmas and Happy Holidays,

Jonathan Vos Post