A231774 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 + 2).

2, 1, 5, 6, 2, 9, 19, 13, 3, 29, 72, 69, 30, 5, 65, 213, 278, 182, 60, 8, 181, 682, 1084, 928, 451, 118, 13, 441, 1975, 3795, 4065, 2625, 1023, 223, 21, 1165, 5868, 13015, 16590, 13290, 6852, 2221, 414, 34, 2929, 16697, 42404, 63020, 60435, 38799, 16682

Offset

1,1

Comments

Sum of numbers in row n: A002534(n). Left edge: A006131. Right edge: A000045 (Fibonacci numbers).

Links

Table of n, a(n) for n=1..51.

Example

First 3 rows:
2 ... 1
5 ... 6 .... 2
9 ... 19 ... 13 ... 3
First 3 polynomials:  2 + x, 5 + 6*x + 2*x^2, 9 + 19*x + 13*x^2 + 3*x^3.

Mathematica

t[n_] := t[n] = Table[(x + 1)/(x + 2), {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, A231775, A000045.

Sequence in context: A078123 A342968 A323312 * A209170 A231732 A185384

Adjacent sequences: A231771 A231772 A231773 * A231775 A231776 A231777

Keyword

nonn,tabf

Author

Clark Kimberling, Nov 13 2013

Status

approved