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A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!. 12
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Previous name was: Decimal expansion of the constant 1 + 1/3! + 1/6! + 1/9! + ... = 1.16805 83133 75918 ... .
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.
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
LINKS
D. Bowman and J. Mc Laughlin, Polynomial continued fractions, arXiv:1812.08251 [math.NT], 2018; Acta Arith. 103 (2002), no. 4, 329-342.
Michael Penn, Two sum identities, YouTube video, 2020.
FORMULA
Equals (exp(1) + exp(w) + exp(w^2))/3, where w = exp(2*Pi*i/3).
A143819 + A143820 + A143821 = exp(1).
Equals 1/3 * (e + 2 * cos(sqrt(3)/2) / sqrt(e)). - Bernard Schott, Mar 01 2020
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
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
EXAMPLE
1.168058313375918525516256929611144747716933295113292516385891232685...
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.
MATHEMATICA
RealDigits[ N[ 1/3*(2*Cos[Sqrt[3]/2]/Sqrt[E] + E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)
With[{nn=120}, RealDigits[N[Total[Table[1/(3n)!, {n, nn}]]+1, nn], 10, nn][[1]]] (* Harvey P. Dale, Apr 20 2013 *)
PROG
(PARI) suminf(k=0, 1/(3*k)!) \\ Michel Marcus, Feb 21 2016
CROSSREFS
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.
Sequence in context: A360500 A308314 A097909 * A021599 A072553 A089991
KEYWORD
easy,nonn,cons
AUTHOR
Peter Bala, Sep 03 2008
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 05 2009
New name from Bernard Schott, Mar 02 2020
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)