A143625 Decimal expansion of the constant E_3(0) := Sum_{n >= 0} (-1)^floor(n/3)/n! = 1 + 1/1! + 1/2! - 1/3! - 1/4! - 1/5! + + + - - - ... .

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

Offset: 1

Author

Peter Bala, Aug 30 2008

Keywords

• cons
• easy
• nonn

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(1) and E_3(2) are given in A143626 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.

Examples

			2.284942382409635208999050...
		

References

• Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.

Crossrefs

Cf. A143623, A143624, A143626, A143627, A143628, A143629, A143630.

Programs

Mathematica

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

Extensions

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