OFFSET
1,3
COMMENTS
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(2) - R(1); the decimal expansions of R(0) = 1 + 1/3! + 1/6! + ... and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143819 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).
R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
========================================
| linear combination of
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
...
LINKS
Michael I. Shamos, A catalog of the real numbers, (2011). See p. 76.
FORMULA
Equals (exp(1) + w^2*exp(w) + w*exp(w^2))/3, where w = exp(2*Pi*i/3).
Equals Sum_{n>=0} 1/(3*n+1)!. - Michal Paulovic, Aug 20 2023
Continued fraction: 1 + 1/(24 - 24/(211 - 210/(721 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (3*n - 1)*(3*n)*(3*n + 1) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024
EXAMPLE
1.041865355098909...
MAPLE
Digits:=101: evalf(sum(1/(3*n+1)!, n=0..infinity)); # Michal Paulovic, Aug 20 2023
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 *)
PROG
(PARI) suminf(n=0, 1/(3*n+1)!) \\ Michel Marcus, Aug 20 2023
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Sep 03 2008
EXTENSIONS
Offset corrected by R. J. Mathar, Feb 05 2009
STATUS
approved