A230000 - OEIS

A230000

Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(k)*x^k which is the numerator of the n-th convergent of the continued fraction [1, 1/x, 1/x^2, ... ,1/x^n].

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 1, 2, 1

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OFFSET

0,36

COMMENTS

In the Name section, k = n(n+1)/2. For the denominator polynomials, see A230001. Conjecture: every nonnegative integer occurs infinitely many times.

LINKS

Table of n, a(n) for n=0..85.

FORMULA

Write the numerator polynomials as u(0), u(1), u(2), ... and the denominator polynomials as v(0), v(1), v(2),... Let p(0) = 1, q(0) = 1; p(1) = (1 + x)/x; q(1) = 1/x; p(n ) = p(n-1)/x^n + p(n-2), q(n) = q(n-1)/x^n + q(n-2). Then u(n)/v(n) = p(n)/q(n) for n>=0.

EXAMPLE

The first 7 rows:

1 . . . . . . . . . . . . polynomial u(0) = 1

1 1 . . . . . . . . . . . polynomial u(1) = 1 + x

1 1 0 1 . . . . . . . . . u(2) = 1 + x + x^3

1 1 0 1 0 1 1

1 1 0 1 0 1 1 1 1 0 1

1 1 0 1 0 1 1 1 1 1 2 0 1 0 1 1

1 1 0 1 0 1 1 1 1 1 2 1 2 0 2 1 1 1 1 1 0 1

MATHEMATICA

t[n_] := t[n] = Table[1/x^k, {k, 0, n}];

b = Table[Factor[Convergents[t[n]]], {n, 0, 10}];