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A230001 Triangular array read by rows: row n shows the coefficients of the polynomial v(n) = d(0) + d(1)*x + ... + d(k)*x^k which is the denominator of the n-th convergent of the continued fraction [1, 1/x, 1/x^2, ... ,1/x^n]. 1
1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 0, 2, 0, 2, 0, 1, 1, 1, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,53

COMMENTS

In the Name section, k = n(n+1)/2 if n is even, and k = (n-1)(n+2)/2 if n is odd.  For the numerator polynomials, see A230000.  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 v(0) = 1

1 . . . . . . . . . . . . polynomial v(1) = 1

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

1 0 0 1 0 1 . . . . . . . v(3) = 1 + x^3 + x^5

1 0 0 1 0 1 0 1 0 0 1

1 0 0 1 0 1 0 1 0 1 1 0 1 0 1

MATHEMATICA

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

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

p[x_, n_] := p[x, n] = Last[Expand[Numerator[b]]][[n]];

u = Table[p[x, n], {n, 1, 10}]

v = CoefficientList[u, x]

Flatten[v]

CROSSREFS

Cf. A230000.

Sequence in context: A324832 A035203 A173920 * A070100 A070095 A060951

Adjacent sequences:  A229998 A229999 A230000 * A230002 A230003 A230004

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Oct 11 2013

STATUS

approved

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Last modified December 4 18:08 EST 2021. Contains 349526 sequences. (Running on oeis4.)