A060997 Decimal representation of continued fraction 1, 2, 3, 4, 5, 6, 7, ...

1, 4, 3, 3, 1, 2, 7, 4, 2, 6, 7, 2, 2, 3, 1, 1, 7, 5, 8, 3, 1, 7, 1, 8, 3, 4, 5, 5, 7, 7, 5, 9, 9, 1, 8, 2, 0, 4, 3, 1, 5, 1, 2, 7, 6, 7, 9, 0, 5, 9, 8, 0, 5, 2, 3, 4, 3, 4, 4, 2, 8, 6, 3, 6, 3, 9, 4, 3, 0, 9, 1, 8, 3, 2, 5, 4, 1, 7, 2, 9, 0, 0, 1, 3, 6, 5, 0, 3, 7, 2, 6, 4, 3, 5, 7, 8, 6, 1, 1, 4, 6, 5, 9, 5, 0

Offset: 1

Author

Robert G. Wilson v, May 14 2001

Keywords

• cons
• easy
• nonn

Comments

The value of this continued fraction is the ratio of two Bessel functions: BesselI(0,2)/BesselI(1,2) = A070910/A096789. Or, equivalently, to the ratio of the sums: Sum_{n>=0} 1/(n!n!) and Sum_{n>=0} n/(n!n!). - Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

1.43312...=[1,2,3,4,5,...] = shape of a rectangle which partitions into n squares at stage n; i.e., this is an example of the match between the continued fraction of a number r and a rectangle having shape r. See A188640. - Clark Kimberling, Apr 09 2011

Examples

			1.433127426722311758317183455775...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000 (corrected by _Sean A. Irvine_, Apr 29 2022)
• J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
• J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]

Crossrefs

Cf. A052119, A001053.

Programs

Mathematica

With[{nn = 110}, RealDigits[FromContinuedFraction[Range[nn]], 10, nn][[1]]]
(* Or *) RealDigits[ BesselI[0, 2] / BesselI[1, 2], 10, 110] [[1]]
(* Or *) RealDigits[ Sum[1/(n!n!), {n, 0, Infinity}] / Sum[n/(n!n!), {n, 0, Infinity}], 10, 110] [[1]]

Maxima

set_display('none)$fpprec:100$bfloat(cfdisrep(makelist(x,x,1,1000))); /* Dimitri Papadopoulos, Oct 25 2022 */

PARI

besseli(0,2)/besseli(1,2) \\ Charles R Greathouse IV, Feb 19 2014

Formula

1/A052119.