`k\h` | `(A167809) 2` | `(A167810) 3` | `(A167811) 4` | `(A167812) 5` | `(A167813) 6` | `(A167814) 7` |

- One can quantify the size of the search space and its growth rate, and assess the limits of practicality of a brute force solver.
- Having some terms is better than none when you're trying to define a tight asymptotic bound, or better yet, an exact formula.
- More practically, these are the benchmark numbers to compare to once you start pruning the search tree.
- It's always useful to have these sequences on record since they may unexpectedly pop up elsewhere, or be in some other way related to other sequences in the encyclopedia.

`h = 2`

is related to Goldbach's conjecture somehow. In fact, A167809 is essentially a duplicate of A008932, which has an impressively short PARI code attached. I wish I know what it does, see if it can be generalized. In fact, there are a lot of important results for `h = 2`

, of which any generalization would be a huge breakthrough.(update) I've figured what the PARI code does, and yes, it is quite trivially generalizable to

`h > 2`

. This is a straightly mathematical approach to the postage stamp problem.
To illustrate, let's look at the case when `h = 3`

and the basis is `{ 0 } U { 1, 2, 5 }`

.
> sort(expand((x^0 + x^1 + x^2 + x^5)^3)); 15 12 11 10 9 8 7 6 5 4 3 x + 3 x + 3 x + 3 x + 3 x + 6 x + 9 x + 7 x + 6 x + 6 x + 7 x 2 + 6 x + 3 x + 1A term

`c[i]*x^i`

means that the number `i`

is obtainable in `c[i]`

different ways as the sum of exactly `h`

numbers in the basis (including the introduced `0`

), allowing permutations (e.g. `9x^7 ->`

7 = 0+2+5 = 0+5+2 = 2+0+5 = 2+5+0 = 5+0+2 = 5+2+0 = 1+1+5 = 1+5+1 = 5+1+1).
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