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A143821
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Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .
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6
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5, 0, 8, 3, 5, 8, 1, 5, 9, 9, 8, 4, 2, 1, 6, 8, 6, 3, 5, 4, 2, 6, 9, 3, 9, 2, 6, 7, 1, 9, 9, 9, 0, 3, 6, 2, 3, 4, 3, 2, 3, 0, 2, 2, 6, 8, 6, 2, 5, 0, 3, 5, 9, 9, 0, 3, 5, 3, 3, 7, 1, 3, 9, 6, 1, 5, 4, 1, 1, 4, 4, 2, 7, 1, 9, 2, 6, 7, 9, 9, 3, 1, 8, 7, 6, 4, 7, 0, 2, 4, 0, 0, 9, 5, 4, 6, 5, 8, 2, 5
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OFFSET
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0,1
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COMMENTS
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Define a sequence of real numbers R(n) by R(n) := sum {k = 0..inf} (3k)^n/(3k)! for n = 0,1,2... . This constant is R(1); the decimal expansions of R(0) = 1 + 1/3!+ 1/6! + 1/9! + ... and R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... may be found in A143819 and A143820. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + sum {i = 0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1). Some examples are given below.
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LINKS
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Table of n, a(n) for n=0..99.
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FORMULA
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Constant = (exp(1) + w*exp(w) + w^2*exp(w^2))/3, where w = exp(2*Pi*i/3). A143819 + A143820 + A143821 = exp(1).
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EXAMPLE
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R(n) as a linear combination of R(0),R(1)
and R(2) - R(1).
=======================================
..R(n)..|.....R(0).....R(1)...R(2)-R(1)
=======================================
..R(3)..|.......1........1........3....
..R(4)..|.......6........2........7....
..R(5)..|......25.......11.......16....
..R(6)..|......91.......66.......46....
..R(7)..|.....322......352......203....
..R(8)..|....1232.....1730.....1178....
..R(9)..|....5672.....8233.....7242....
..R(10).|...32202....39987....43786....
...
The column entries are from A143815, A143816 and A143817.
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MATHEMATICA
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RealDigits[ N[ -((Cos[Sqrt[3]/2] - E^(3/2) + Sqrt[3]*Sin[Sqrt[3]/2])/(3*Sqrt[E])), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)
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CROSSREFS
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A073742, A073743, A143815, A143816, A143817, A143818, A143819, A143820.
Sequence in context: A078119 A085998 A094886 * A099219 A201288 A011441
Adjacent sequences: A143818 A143819 A143820 * A143822 A143823 A143824
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KEYWORD
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cons,easy,nonn
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AUTHOR
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Peter Bala, Sep 03 2008
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EXTENSIONS
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Offset corrected by R. J. Mathar, Feb 05 2009
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STATUS
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approved
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