A060997 - OEIS

%I #48 Apr 06 2024 09:03:11

%S 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,

%T 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,

%U 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

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

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A060997/b060997.txt">Table of n, a(n) for n = 1..5000</a> (corrected by _Sean A. Irvine_, Apr 29 2022)

%H J. M. Borwein, <a href="https://carmamaths.org/resources/jon/OEIStalk.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.

%H J. M. Borwein, <a href="/A060997/a060997.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]

%F 1/A052119.

%e 1.433127426722311758317183455775...

%t With[{nn = 110}, RealDigits[FromContinuedFraction[Range[nn]], 10, nn][[1]]]

%t (* Or *) RealDigits[ BesselI[0, 2] / BesselI[1, 2], 10, 110] [[1]]

%t (* Or *) RealDigits[ Sum[1/(n!n!), {n, 0, Infinity}] / Sum[n/(n!n!), {n, 0, Infinity}], 10, 110] [[1]]

%o (PARI) besseli(0,2)/besseli(1,2) \\ _Charles R Greathouse IV_, Feb 19 2014

%o (Maxima) set_display('none)$fpprec:100$bfloat(cfdisrep(makelist(x,x,1,1000))); /* _Dimitri Papadopoulos_, Oct 25 2022 */

%Y Cf. A052119, A001053.

%K cons,easy,nonn

%O 1,2

%A _Robert G. Wilson v_, May 14 2001