A143630 -id:A143630 - cp's OEIS frontend

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

Peter Bala, Aug 30 2008

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.

			1.3015594959829796028430427

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

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

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

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 ... .

Original entry on oeis.org

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

Peter Bala, Aug 30 2008

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.

			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.

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

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 *)

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

cp's OEIS Frontend

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! + + + - - - ... .

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