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A060997
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Decimal representation of continued fraction 1, 2, 3, 4, 5, 6, 7, ...
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25
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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1.433127426722311758317183455775...
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MATHEMATICA
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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]]
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PROG
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(Maxima) set_display('none)$fpprec:100$bfloat(cfdisrep(makelist(x, x, 1, 1000))); /* Dimitri Papadopoulos, Oct 25 2022 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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