A229057 -id:A229057

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A033165

First occurrence of n as a term in the continued fraction for zeta(3).

+10
3

1, 12, 25, 2, 64, 27, 17, 140, 10, 119, 21, 239, 175, 78, 181, 46, 200, 4, 83, 619, 753, 412, 177, 197, 414, 138, 146, 561, 233, 29, 2276, 1549, 660, 889, 298, 1040, 2279, 322, 1274, 1882, 345, 2926, 673, 254, 1961, 1542, 1681, 296, 5423, 2423, 2557, 228

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Incorrectly indexed version of A229057.

LINKS

Table of n, a(n) for n=1..52.

Eric Weisstein's World of Mathematics, Apery's Constant Continued Fraction

FORMULA

a(n) = 1 + A229057(n).

MATHEMATICA

With[{cfz3 = ContinuedFraction[Zeta[3], 6000]}, Flatten[Table[Position[cfz3, n, 1, 1], {n, 60}]]] (* Harvey P. Dale, Nov 11 2012 *)

PROG

(PARI) /* 1500 precision digits */ v=contfrac(zeta(3)); a(n)=if(n<0, 0, s=1; while(abs(n-component(v, s))>0, s++); s)

CROSSREFS

Cf. A013631, A032523, A229057.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Randall L Rathbun, Feb 03 2002

STATUS

approved

A249303

Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

+10
1

1, 0, 1, -1, 2, -1, 1, 1, 0, -2, 3, 1, -4, 3, 1, 1, -2, -2, 4, 0, 3, -9, 6, 1, -1, 6, -9, 0, 5, -1, 3, 3, -15, 10, 1, 0, -4, 18, -24, 5, 6, 1, -8, 18, -6, -20, 15, 1, 1, -4, -4, 36, -49, 14, 7, 0, 5, -30, 60, -35, -21, 21, 1, -1, 10, -30, 20, 50, -84, 28, 8

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = 1 + (x - 1)/f(n-1,x), where f(0,x) = 1.

Every row sum is 1. The first column is purely periodic with period (1,0,-1,-1,0,1).

Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime - 2). More generally, if c is arbitrary and f(n,x) = 1 + (x + c)/f(n-1,x), where f(x,0) = 1, then p(n,x) is irreducible if and only if n is a (prime - 2).

LINKS

Clark Kimberling, Rows 0..100, flattened

EXAMPLE

f(0,x) = 1/1, so that p(0,x) = 1

f(1,x) = x/1, so that p(1,x) = x;

f(2,x) = (-1 + 2 x)/x, so that p(2,x) = -1 + 2 x.

First 6 rows of the triangle of coefficients:

... 1

... 0 ... 1

.. -1 ... 2

.. -1 ... 1 ... 1

... 0 .. -2 ... 3

... 1 .. -4 ... 3 ... 1

MATHEMATICA

z = 20; f[n_, x_] := 1 + (x - 1)/f[n - 1, x]; f[0, x_] = 1;

t = Table[Factor[f[n, x]], {n, 0, z}]

u = Numerator[t]

TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]] (* A249303 array *)

v = Flatten[CoefficientList[u, x]] (* A249303 *)

CROSSREFS

Cf. A128100, A229057, A229074.

KEYWORD

tabf,sign,easy

AUTHOR

Clark Kimberling, Oct 24 2014

STATUS

approved

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