A128384 a(n) = numerator of r(n): r(n) is such that the continued fraction (of rational terms) [r(1);r(2),...,r(n)] = b(n) for every positive integer n, where b(1) = 1 and b(n+1) = 1 + 1/b(n)^2 for.every positive integer n.

1, 1, 1, 9, 91, 1110627, 2340346013551, 1153990596161360976412397283, 4046008078695782327150934410204261346038863974869654053

Offset

1,4

Comments

b(n) = A076725(n)/A076725(n-1)^2. The limit, as n -> infinity, of r(n)*r(n+1) = (2 /x^3) + (x^3 /2) - 2, where x is the real root of x^3 -x^2 -1 = 0. (This limit result needs some checking.)

a(10) has 113 digits. - Michel Marcus, Jan 13 2014

Links

Table of n, a(n) for n=1..9.

Example

{r(n)}: 1, 1, 1/3, 9/13, 91/289,...
b(4) = 41/25 = 1 + 1/(1 + 1/(1/3 + 13/9)).
And b(5) = 2306/1681 = 1 + 1/(1 + 1/(1/3 + 1/(9/13 + 289/91))).

Prog

(PARI) lista(nn) = {kill(n); kill(m); nn1 = nn + 1; v = vector(nn1); v[1] = 1; v[2] = 1; for (i=3, nn1, v[i] = v[i-1]^2 + v[i-2]^4; ); b = vector(nn1, i, if (i==1, 1, v[i]/v[i-1]^2)); r = vector(nn); r[1] = b[1]; for (i=2, nn, pol = (n/m); forstep (k = i-1, 1, -1, pol = 1/pol + r[k]; ); mat = matrix(2, 2); mat[1, 1] = polcoeff( polcoeff( numerator(pol), 1, n), 0, m); mat[1, 2] = polcoeff( polcoeff( numerator(pol), 1, m), 0, n); mat[2, 1] = polcoeff( polcoeff( denominator(pol), 1, n), 0, m); mat[2, 2] = polcoeff( polcoeff( denominator(pol), 1, m), 0, n); col = vector(2); col[1] = numerator(b[i+1]); col[2] = denominator(b[i+1]); vres = matsolve(mat, col~); r[i] = vres[1]/vres[2]; ); r; } \\ Michel Marcus, Jan 12 2014

Crossrefs

Cf. A128385, A076725.

Sequence in context: A034666 A022505 A225162 * A164913 A007403 A015587

Adjacent sequences: A128381 A128382 A128383 * A128385 A128386 A128387

Keyword

frac,nonn

Author

Leroy Quet, Feb 28 2007

Extensions

More terms from Michel Marcus, Jan 12 2014

Status

approved