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

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

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

Table of n, a(n) for n=1..101.

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.

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.

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. A073742, A073743, A143815, A143816, A143817, A143818, A143820, A143821.

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 * A372605 A021599 A072553

Adjacent sequences: A143816 A143817 A143818 * A143820 A143821 A143822

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