OFFSET
0,1
COMMENTS
Define a sequence of real numbers R(n) by R(n) := Sum_{k >= 0} (3*k)^n/(3*k)! 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.
LINKS
Michael I. Shamos, A catalog of the real numbers, (2011). See p. 460.
FORMULA
Constant = (exp(1) + w*exp(w) + w^2*exp(w^2))/3, where w = exp(2*Pi*i/3). A143819 + A143820 + A143821 = exp(1).
Continued faction: 1/(2 - 2/(61 - 60/(337 - 336/(991 - ... - P(n-1)/((P(n) + 1) - ... ))))), where P(n) = (3*n)*(3*n + 1)*(3*n + 2) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024
EXAMPLE
R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
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..R(n)..|.....R(0).....R(1)...R(2)-R(1)
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..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....
...
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Sep 03 2008
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved