A143820 Decimal expansion of the constant 1/1! + 1/4! + 1/7! + ...

1, 0, 4, 1, 8, 6, 5, 3, 5, 5, 0, 9, 8, 9, 0, 9, 8, 4, 6, 3, 0, 1, 3, 3, 6, 6, 1, 5, 0, 2, 1, 5, 2, 7, 3, 8, 7, 6, 9, 7, 0, 8, 3, 5, 7, 1, 7, 2, 4, 1, 6, 3, 4, 5, 9, 5, 4, 5, 7, 3, 9, 2, 5, 5, 4, 2, 3, 5, 5, 1, 7, 4, 1, 1, 6, 1, 0, 7, 4, 0, 2, 9, 5, 9, 2, 8, 6, 2, 6, 7, 3, 9, 3, 0, 1, 0, 0, 6, 5, 5, 2

Offset: 1

Author

Peter Bala, Sep 03 2008

Keywords

• cons
• easy
• nonn

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

...

The column entries are from A143815, A143816 and A143817.

The Abraham Ungar 1982 article defines H_{n,r}(z) = Sum_{k>=0} z^(nk+r)/(nk+r)! as equation (1). The constant is H_{3,1}(1). In equation (13) H_{3,1}(x) = (exp(x) + 2 * exp(-x/2) * cos(sqrt(3)/2*x - 2*Pi/3))/3. In equation (12) the expression H_{3,1}(x) = (e^x + q_2 e^{q_1 x} + q_1 e^{q_2 x})/3 where q_1 = (-1 + I sqrt(3))/2 and q_2 = (-1 - I sqrt(3))/2 is given for H_{3,2}(x) instead. - Michael Somos, Nov 01 2024

Examples

			1.041865355098909...
		

Links

• Michael I. Shamos, A catalog of the real numbers, (2011). See p. 76.
• Abraham Ungar, Generalized Hyperbolic Functions, The American Mathematical Monthly, Volume 89, 1982, 688-691.

Crossrefs

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143819, A143821, A346441.

Programs

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 *)

PARI

suminf(n=0,1/(3*n+1)!) \\ Michel Marcus, Aug 20 2023

Formula

Equals (exp(1) + w^2*exp(w) + w*exp(w^2))/3, where w = exp(2*Pi*i/3).

A143819 + A143820 + A143821 = exp(1).

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

Equals (exp(1) + 2*exp(-1/2)*cos(sqrt(3)/2-2*Pi/3))/3. [Ungar, p.690] - Michael Somos, Nov 01 2024

Extensions

Offset corrected by R. J. Mathar, Feb 05 2009