A143627 - OEIS

%I #7 Nov 08 2012 11:22:17

%S 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,

%T 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,

%U 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

%N 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 ... .

%C 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 E_3(n) as linear combination of E_3(i),

%e i = 0..2.

%e =======================================

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

%e =======================================

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

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

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

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

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

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

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

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

%e ...

%e The columns are A143628, A143629 and A143630.

%t 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 *)

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

%K cons,easy,nonn

%O 0,1

%A _Peter Bala_, Aug 30 2008

%E Offset corrected by _R. J. Mathar_, Feb 05 2009