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A143821 Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .

%I A143821 #28 Jan 19 2025 11:04:37
%S A143821 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,
%T A143821 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,
%U A143821 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
%N A143821 Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .
%C A143821 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.
%H A143821 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2011). See p. 460.
%F A143821 Constant = (exp(1) + w*exp(w) + w^2*exp(w^2))/3, where w = exp(2*Pi*i/3). A143819 + A143820 + A143821 = exp(1).
%F A143821 Continued fraction: 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
%e A143821 R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
%e A143821 =======================================
%e A143821 ..R(n)..|.....R(0).....R(1)...R(2)-R(1)
%e A143821 =======================================
%e A143821 ..R(3)..|.......1........1........3....
%e A143821 ..R(4)..|.......6........2........7....
%e A143821 ..R(5)..|......25.......11.......16....
%e A143821 ..R(6)..|......91.......66.......46....
%e A143821 ..R(7)..|.....322......352......203....
%e A143821 ..R(8)..|....1232.....1730.....1178....
%e A143821 ..R(9)..|....5672.....8233.....7242....
%e A143821 ..R(10).|...32202....39987....43786....
%e A143821 ...
%e A143821 The column entries are from A143815, A143816 and A143817.
%t A143821 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 *)
%Y A143821 Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143819, A143820, A346441.
%K A143821 cons,easy,nonn
%O A143821 0,1
%A A143821 _Peter Bala_, Sep 03 2008
%E A143821 Offset corrected by _R. J. Mathar_, Feb 05 2009