cp's OEIS Frontend

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.

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A384383 Number of polynomials with a shortest addition-multiplication-composition chain of length n, starting with 1 and x.

Original entry on oeis.org

2, 4, 14, 73, 586, 7250
Offset: 0

Views

Author

Pontus von Brömssen, Jun 01 2025

Keywords

Comments

An addition-multiplication-composition chain for the polynomial p(x) is a finite sequence of polynomials, starting with 1, x and ending with p(x), in which each element except 1 and x equals q(x)+r(x), q(x)*r(x), or q(r(x)) for two preceding, not necessarily distinct, elements q(x) and r(x) in the chain. The length of the chain is the number of elements in the chain, excluding 1 and x.

Examples

			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).

Crossrefs

Cf. A382928, A383331 (addition only), A384382 (addition and multiplication), A384386, A384482 (addition and composition).

A384382 Number of polynomials with a shortest addition-multiplication chain of length n, starting with 1 and x.

Original entry on oeis.org

2, 4, 14, 62, 350, 2517, 22918, 259325
Offset: 0

Views

Author

Pontus von Brömssen, Jun 01 2025

Keywords

Comments

An addition-multiplication chain for the polynomial p(x) is a finite sequence of polynomials, starting with 1, x and ending with p(x), in which each element except 1 and x equals q(x)+r(x) or q(x)*r(x) for two preceding, not necessarily distinct, elements q(x) and r(x) in the chain. The length of the chain is the number of elements in the chain, excluding 1 and x.

Examples

			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.

Crossrefs

Cf. A382928, A383002, A383331 (addition only), A384383 (addition, multiplication, and composition), A384482 (addition and composition).
Showing 1-2 of 2 results.

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