Polynomial reduction: an introduction
We begin with an example. Suppose that p(x) is a polynomial, so that p(x)=(x^2)t(x)+r(x) for some polynomials t(x) and r(x), where r(x) has degree 0 or 1. Replace x^2 by x+1 to get (x+1)t(x)+r(x), which is (x^2)u(x)+v(x) for some u(x) and v(x), where v(x) has degree 0 or 1. Continuing in this manner results in a fixed polynomial w(x) of degree 0 or 1. If p(x)=x^n, then w(x)=x*F(n)+F(n-1), where F=A000045, the sequence of Fibonacci numbers.
In order to generalize, write d(g) for the degree of an arbitrary polynomial g(x), and suppose that p, q, s are polynomials satisfying d(s)<d(q). By the division algorithm, there exists a unique pair t and r of polynomials such that p=q*t+r and d(r)<d(q). Replace q by s to get s*t+r, which is q*u+v for some u and v, where d(v)<d(q). Continue applying q->s in this manner until reaching w such that d(w)<d(q). We call w the reduction of p by q->s.
The coefficients of (reduction of p by q->s) comprise a vector of length d(q)-1, so that a sequence p(n,x) of polynomials begets a sequence of vectors, such as (F(n), F(n-1)) in the above example. We are interested in the component sequences (e.g., F(n-1) and F(n)) for various choices of p(n,x).
Following are examples of reduction by x^2->x+1:
n-th Fibonacci p(x) -> A192232+x*A112576
n-th cyclotomic p(x) -> A192233+x*A051258
n-th 1st-kind Chebyshev p(x) -> A192234+x*A071101
n-th 2nd-kind Chebyshev p(x) -> A192235+x*A192236
x(x+1)(x+2)...(x+n-1) -> A192238+x*A192239
(x+1)^n -> A001519+x*A001906
(x^2+x+1)^n -> A154626+x*A087635
(x+2)^n -> A020876+x*A030191
(x+3)^n -> A192240+x*A099453
Suppose that b=(b(0), b(1),...) is a sequence, and let p(n,x)=b(0)+b(1)x+b(2)x^2+...+b(n)x^n. We define (reduction of sequence b by q->s) to be the vector given by (reduction of p(n,x) by q->s), with components in the order of powers, from 0 up to d(q)-1. For k=0,1,...,d(q)-1, we then have the "k-sequence of (reduction of sequence b by q->s)". Continuing the example, if b is the sequence given by b(k)=1 if k=n and b(k)=0 otherwise, then the 0-sequence of (reduction of b by x^2->x+1) is (F(n-1)), and the 1-sequence is (F(n)).
For selected sequences b, here are the 0-sequences and 1-sequences of (reduction of b by x^2->x+1):
b=A000045, Fibonacci sequence (1,1,2,3,5,8,...) yields
0-sequence A166536 and 1-sequence A064831.
0-sequence A166516 and 1-sequence A001654.
b=A000027, natural number sequence (1,2,3,4,...) yields
0-sequence A190062 and 1-sequence A122491.
b=A000032, Lucas sequence (1,3,4,7,11,...) yields
0-sequence A192243 and 1-sequence A192068.
b=(A000217, triangular sequence (1,3,6,10,...) yields
0-sequence A192244 and 1-sequence A192245.
b=(A000290, squares sequence (1,4,9,16,...) yields
0-sequence A192254 and 1-sequence A192255.
More examples: A192245-A192257.
(1) If s(n,x)=(reduction of x^n by q->s) and
(reduction of p by q->s)=p(0)s(n,x)+p(1)s(n-1,x)
+...+p(n-1)s(1,x)+p(n)s(0,x). See A192744.
(2) For any polynomial p(x), let P(x)=(reduction of p(x)
by q->s). Then P(r)=p(r) for each zero r of
q(x)-s(x). In particular, if q(x)=x^2 and s(x)=x+1,
then P(r)=p(r) if r=(1+sqrt(5))/2 (golden ratio) or