This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
An example of a polynomial for which composition is necessary to obtain the shortest chain is 9*x, with the chain (1, x,) 2*x, 3*x, 9*x. (9*x is the composition of 3*x with itself.) So 9*x is one of the 11 polynomials counted by a(3) but not by A384382(3).
n | a(n) | pairs (x,y) with x <= y, x+y = a(n), and shortest chain length n ---+------+----------------------------------------------------------------- 0 | 1 | (0,1) 1 | 2 | (0,2), (1,1) 2 | 3 | (0,3), (1,2) 3 | 4 | (1,3) 4 | 6 | (1,5) 5 | 8 | (1,7) 6 | 12 | (1,11) 7 | 20 | (1,19), (3,17) 8 | 29 | (6,23) 9 | 44 | (7,37) 10 | 70 | (11,59) 11 | 104 | (15,89)
Triangle begins: n\k| 0 1 2 3 4 5 6 7 8 9 10 ---+-------------------------------- 0 | 0 1 | 0 1 2 | 1 2 2 3 | 2 3 3 3 4 | 2 3 3 4 3 5 | 3 4 4 4 4 4 6 | 3 4 4 4 4 5 4 7 | 4 5 5 5 5 5 5 5 8 | 3 4 4 5 4 5 5 6 4 9 | 4 5 5 5 5 5 5 6 5 5 10 | 4 5 5 5 5 5 5 6 5 6 5 A shortest addition chain for (11,7) is [(1,0), (0,1),] (1,1), (2,1), (4,2), (5,3), (10,6), (11,7) of length T(11,7) = 6.
Array begins:
n\k| 0 1 2 3 4 5 6 7
---+-----------------------------------
1 | 1 1 2 3 5 9 15 26
2 | 2 3 7 16 37 91 229 585
3 | 3 6 16 46 134 411 1319 4368
4 | 4 10 30 101 349 1264 4817 19131
5 | 5 15 50 190 751 3106 13532 61748
6 | 6 21 77 323 1426 6586 32035 163594
There are 12 triples of nondecreasing nonnegative integers with a shortest addition chain of length 3. Counting also the permutations of these, we get T(3,3) = 46:
(0, 0, 5): 3
(0, 0, 6): 3
(0, 0, 8): 3
(0, 1, 3): 6
(0, 1, 4): 6
(0, 2, 3): 6
(0, 2, 4): 6
(0, 3, 3): 3
(0, 4, 4): 3
(1, 1, 2): 3
(1, 2, 2): 3
(2, 2, 2): 1
Total: 46
a(0) = 2 because 1 and x are considered to have chains of length 0. a(1) = 4 because the 4 polynomials 2, x+1, 2*x, and x^2 have chains of length 1. a(2) = 14 because the 14 polynomials 3, 4, x+2, 2*x+1, 2*x+2, 3*x, 4*x, x^2+1, x^2+x, x^2+2*x+1, 2*x^2, 4*x^2, x^3, and x^4 have chains of length 2.
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