A143626 Decimal expansion of the constant E_3(1) := Sum_{k >= 0} (-1)^floor(k/3)*k/k! = 1/1! + 2/2! - 3/3! - 4/4! - 5/5! + + + - - - ... .

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

Offset

1,2

Comments

Define E_3(n) = Sum_{k >= 0} (-1)^floor(k/3)*k^n/k! = 0^n/0! + 1^n/1! + 2^n/2! - 3^n/3! - 4^n/4! - 5^n/5! + + + - - - ... for n = 0,1,2,... . It is easy to see that E_3(n+3) = 3*E_3(n+2) - 2*E_3(n+1) - Sum_{i = 0..n} 3^i*binomial(n,i) * E_3(n-i) for n >= 0. Thus E_3(n) is an integral linear combination of E_3(0), E_3(1) and E_3(2). See the examples below.

The decimal expansions of E_3(0) and E_3(2) are given in A143635 and A143627. Compare with A143623 and A143624.

E_3(n) as linear combination of E_3(i), i = 0..2.

=======================================

..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)

=======================================

..E_3(3)..|.....-1.......-2........3...

..E_3(4)..|.....-6.......-7........7...

..E_3(5)..|....-25......-23.......14...

..E_3(6)..|....-89......-80.......16...

..E_3(7)..|...-280.....-271......-77...

..E_3(8)..|...-700.....-750.....-922...

..E_3(9)..|...-380.....-647....-6660...

..E_3(10).|..13452....13039...-41264...

...

The columns are A143628, A143629 and A143630.

Links

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

Example

1.3015594959829796028430427

Mathematica

RealDigits[ N[ (4*E^(3/2)*Cos[Sqrt[3]/2] - 1)/(3*E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Crossrefs

Cf. A143623, A143624, A143625, A143627, A143628, A143629, A143630.

Sequence in context: A186363 A356777 A254295 * A034261 A046778 A119925

Adjacent sequences: A143623 A143624 A143625 * A143627 A143628 A143629

Keyword

cons,easy,nonn

Author

Peter Bala, Aug 30 2008

Extensions

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

Status

approved