A229057 -id:A229057 - cp's OEIS frontend

A033165 First occurrence of n as a term in the continued fraction for zeta(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

Offset: 1

N. J. A. Sloane

nonn

Incorrectly indexed version of A229057.

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

Cf. A013631, A032523, A229057.

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

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

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

More terms from Randall L Rathbun, Feb 03 2002

A249303 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 24 2014

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).

			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

Clark Kimberling, Rows 0..100, flattened

Cf. A128100, A229057, A229074.

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 *)

cp's OEIS Frontend

A033165 First occurrence of n as a term in the continued fraction for zeta(3).

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