STATUS
editing
approved
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Revision History for A230000
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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].
(history; published version)
(history; published version)
NAME
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 nth n-th convergent of the continued fraction [1, 1/x, 1/x^2, ... ,1/x^n].
STATUS
approved
editing
STATUS
editing
approved
NAME
ArrayTriangular 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 nth convergent of the continued fraction [1, 1/x, 1/x^2, ... ,1/x^n].
STATUS
proposed
editing
STATUS
editing
proposed
NAME
allocated for Clark KimberlingArray: 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 nth convergent of the continued fraction [1, 1/x, 1/x^2, ... ,1/x^n].
DATA
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
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.
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}];
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. A230001.
KEYWORD
allocated
nonn,tabf
AUTHOR
Clark Kimberling, Oct 11 2013
STATUS
approved
editing
NAME
allocated for Clark Kimberling
KEYWORD
allocated
STATUS
approved