A231728 Triangular array read by rows: row n shows the coefficients of the polynomial u(n) = c(0) + c(1)*x + ... + c(2n)*x^(2n) which is the numerator of the n-th convergent of the continued fraction [k, k, k, ... ], where k = (x^2 + 1)/(x + 1).

1, 0, 1, 2, 2, 3, 0, 1, 3, 4, 7, 4, 5, 0, 1, 5, 10, 19, 16, 16, 6, 7, 0, 1, 8, 20, 42, 48, 55, 36, 29, 8, 9, 0, 1, 13, 40, 94, 132, 164, 138, 119, 64, 46, 10, 11, 0, 1, 21, 76, 197, 324, 451, 464, 439, 304, 219, 100, 67, 12, 13, 0, 1, 34, 142, 405, 760, 1170

Offset

1,4

Comments

Sum of numbers in row n: 2*A087206(n). Left edge: A000045 (Fibonacci numbers).

Links

Table of n, a(n) for n=1..68.

Example

First 3 rows:
1 . . . 0 . . . 1
2 . . . 2 . . . 3 . . . 0 . . . 1
3 . . . 4 . . . 7 . . . 4 . . . 5 . . . 0 . . . 1
First 3 polynomials:  1 + x^2, 2 + 2*x + 3*x^2 + x^4.

Mathematica

t[n_] := t[n] = Table[(1 + x^2)/(1 + x), {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, A231729.

Sequence in context: A071479 A257398 A182631 * A303545 A306191 A290125

Adjacent sequences: A231725 A231726 A231727 * A231729 A231730 A231731

Keyword

nonn,tabf

Author

Clark Kimberling, Nov 13 2013

Status

approved