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A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.

%I A143819 #73 Apr 05 2024 04:46:44
%S A143819 1,1,6,8,0,5,8,3,1,3,3,7,5,9,1,8,5,2,5,5,1,6,2,5,6,9,2,9,6,1,1,1,4,4,
%T A143819 7,4,7,7,1,6,9,3,3,2,9,5,1,1,3,2,9,2,5,1,6,3,8,5,8,9,1,2,3,2,6,8,5,1,
%U A143819 1,3,4,4,6,4,7,3,2,0,5,5,7,1,7,9,0,8,7,2,4,8,0,5,8,5,5,1,9,1,8,9,6
%N A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.
%C A143819 Previous name was: Decimal expansion of the constant 1 + 1/3! + 1/6! + 1/9! + ... = 1.16805 83133 75918 ... .
%C A143819 Define a sequence R(n) of real numbers by R(n) := Sum_{k>=0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(0); the decimal expansions of R(2) - R(1) = 1/1! + 1/4! + 1/7! and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143820 and A143821. 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.
%C A143819 Bowman and Mc Laughlin (Corollary 10 with m = -1) give a continued fraction expansion for this constant and deduce the constant is irrational. - _Peter Bala_, Apr 17 2017
%H A143819 D. Bowman and J. Mc Laughlin, <a href="https://doi.org/10.4064/aa103-4-3">Polynomial continued fractions</a>, arXiv:1812.08251 [math.NT], 2018; Acta Arith. 103 (2002), no. 4, 329-342.
%H A143819 Michael Penn, <a href="https://www.youtube.com/watch?v=fHkbSsS5TbI">Two sum identities</a>, YouTube video, 2020.
%H A143819 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. 209.
%F A143819 Equals (exp(1) + exp(w) + exp(w^2))/3, where w = exp(2*Pi*i/3).
%F A143819 A143819 + A143820 + A143821 = exp(1).
%F A143819 Equals 1/3 * (e + 2 * cos(sqrt(3)/2) / sqrt(e)). - _Bernard Schott_, Mar 01 2020
%F A143819 Sum_{k>=0} (-1)^k / (3*k)! = (exp(-1) + 2*exp(1/2)*cos(sqrt(3)/2))/ 3 = 0.83471946857721... - _Vaclav Kotesovec_, Mar 02 2020
%F A143819 Continued fraction: 1 + 1/(6 - 6/(121 - 120/(505 - ... - 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 A143819 1.168058313375918525516256929611144747716933295113292516385891232685...
%e A143819 R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
%e A143819 =======================================
%e A143819   R(n)  |     R(0)     R(1)   R(2)-R(1)
%e A143819 =======================================
%e A143819   R(3)  |       1        1        3
%e A143819   R(4)  |       6        2        7
%e A143819   R(5)  |      25       11       16
%e A143819   R(6)  |      91       66       46
%e A143819   R(7)  |     322      352      203
%e A143819   R(8)  |    1232     1730     1178
%e A143819   R(9)  |    5672     8233     7242
%e A143819   R(10) |   32202    39987    43786
%e A143819   ...
%e A143819 The column entries are from A143815, A143816 and A143817.
%t A143819 RealDigits[ N[ 1/3*(2*Cos[Sqrt[3]/2]/Sqrt[E] + E), 105]][[1]] (* _Jean-François Alcover_, Nov 08 2012 *)
%t A143819 With[{nn=120},RealDigits[N[Total[Table[1/(3n)!,{n,nn}]]+1,nn],10,nn][[1]]] (* _Harvey P. Dale_, Apr 20 2013 *)
%o A143819 (PARI) suminf(k=0, 1/(3*k)!) \\ _Michel Marcus_, Feb 21 2016
%Y A143819 Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143820, A143821.
%Y A143819 Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), this sequence (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.
%K A143819 easy,nonn,cons
%O A143819 1,3
%A A143819 _Peter Bala_, Sep 03 2008
%E A143819 Offset corrected by _R. J. Mathar_, Feb 05 2009
%E A143819 New name from _Bernard Schott_, Mar 02 2020