A231154 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(n)*x^n which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x + 1)/(x - 1).

1, 1, 2, 0, 2, 3, 1, 1, 3, 5, 0, 6, 0, 5, 8, 0, 8, 8, 0, 8, 13, -2, 19, 4, 19, -2, 13, 21, -5, 33, 15, 15, 33, -5, 21, 34, -12, 64, 12, 60, 12, 64, -12, 34, 55, -25, 116, 20, 90, 90, 20, 116, -25, 55, 89, -50, 213, 8, 210, 84, 210, 8, 213, -50, 89

Offset

1,3

Comments

Sum of numbers in row n: 2^n. Left and right edges: A000045 (Fibonacci numbers).

Links

Table of n, a(n) for n=1..65.

Example

First 5 rows:
1 . . . 1
2 . . . 0 . . . 2
3 . . . 1 . . . 1 . . . 3
5 . . . 0 . . . 6 . . . 0 . . . 5
8 . . . 0 . . . 8 . . . 8 . . . 0 . . . 8
First 3 polynomials:  1 + x, 2 + 2*x^2, 3 + x + x^2 + 3*x^3.

Mathematica

t[n_] := t[n] = Table[(x + 1)/(x - 1), {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. A230000, A000045, A231727.

Sequence in context: A114327 A330240 A330237 * A073450 A284592 A071447

Adjacent sequences: A231151 A231152 A231153 * A231155 A231156 A231157

Keyword

sign,tabf

Author

Clark Kimberling, Nov 13 2013

Status

approved