A143627 Decimal expansion of the constant E_3(2) := sum {k = 0.. inf} (-1)^floor(k/3)*k^2/k! = 1/1! + 2^2/2! - 3^2/3! - 4^2/4! - 5^2/5! + + + - - - ... = 0.68605 60507 ... .

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

Offset: 0

Author

Peter Bala, Aug 30 2008

Keywords

• cons
• easy
• nonn

Comments

Define E_3(n) = sum {k = 0..inf} (-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(1) are given in A143625 and A143626. Compare with A143623 and A143624.

Examples

			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.
		

Crossrefs

A143623, A143624, A143625, A143626, A143628, A143629, A143630.

Programs

Mathematica

RealDigits[N[(8/3)*Sqrt[E]*Cos[Sqrt[3]/2] + (1/40)*(HypergeometricPFQ[{}, {7/3, 8/3}, -(1/27)] - 5*HypergeometricPFQ[{}, {5/3, 7/3}, -(1/27)]) - 2*Sqrt[E/3]*Sin[Sqrt[3]/2] - 5/(3*E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Extensions

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