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).
Array begins: n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 ---+-------------------------------------- 0 | 0 0 1 2 2 3 3 4 3 4 4 5 4 1 | 0 1 2 3 3 4 4 5 4 5 5 5 5 2 | 1 2 2 3 3 4 4 4 4 5 5 5 5 3 | 2 3 3 3 4 4 4 5 5 5 5 5 5 4 | 2 3 3 3 3 4 3 4 4 4 4 5 4 5 | 3 4 4 4 4 4 4 4 5 5 5 5 5 6 | 3 4 4 4 4 4 4 5 4 4 5 5 5 7 | 4 5 5 5 5 5 5 5 5 5 5 6 6 8 | 3 4 4 4 4 4 4 4 4 5 4 5 4 9 | 3 4 5 4 4 5 5 5 4 5 5 5 4 10 | 4 5 5 5 5 5 5 5 5 5 5 6 5 11 | 4 5 6 5 5 5 6 6 6 5 5 6 6 12 | 4 5 5 5 5 5 5 5 5 5 5 5 5 For (n,k) = (4,6), the unique shortest chain for 4*x+6 is (1, x,) x+1, 2*x+2, 4*x+6 of length T(4,6) = 3. The last term of the chain is the composition of 2*x+2 with itself. For (n,k) = (6,4), a shortest chain for 6*x+4 is (1, x,) x+1, 2*x+2, 3*x+2, 6*x+4 of length T(6,4) = 4. This chain uses only additions.
n | functions b*x+c with b+c = a(n) and shortest chain of length n --+--------------------------------------------------------------- 0 | 1, x 1 | 2, x+1, 2*x 2 | 3, x+2, 2*x+1, 3*x 3 | 3+x, x+3 4 | 5+x, x+5 5 | 7+x, x+7 6 | 11*x+2 7 | 23*x+1 8 | 7*x+39, 43*x+3 9 | 11*x+87
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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