%I A143625
%S A143625 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,
%T A143625 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,
%U A143625 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
%N A143625 Decimal expansion of the constant E_3(0) := sum {n = 0.. inf} (-1)^floor(n/
               3)/n! = 1 + 1/1! + 1/2! - 1/3! - 1/4! - 1/5! + + + - - - ... = 2.28494 
               23824 ... .
%C A143625 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(1) and E_3(2) are 
               given in A143626 and A143627. Compare with A143623 and A143624.
%e A143625 E_3(n) as linear combination of E_3(i),
%e A143625 i = 0..2.
%e A143625 =======================================
%e A143625 ..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)
%e A143625 =======================================
%e A143625 ..E_3(3)..|.....-1.......-2........3...
%e A143625 ..E_3(4)..|.....-6.......-7........7...
%e A143625 ..E_3(5)..|....-25......-23.......14...
%e A143625 ..E_3(6)..|....-89......-80.......16...
%e A143625 ..E_3(7)..|...-280.....-271......-77...
%e A143625 ..E_3(8)..|...-700.....-750.....-922...
%e A143625 ..E_3(9)..|...-380.....-647....-6660...
%e A143625 ..E_3(10).|..13452....13039...-41264...
%e A143625 ...
%e A143625 The columns are A143628, A143629 and A143630.
%Y A143625 A143623, A143624, A143626, A143627, A143628, A143629 and A143630.
%Y A143625 Sequence in context: A144816 A134812 A144847 this_sequence A003612 A103839 
               A135727
%Y A143625 Adjacent sequences: A143622 A143623 A143624 this_sequence A143626 A143627 
               A143628
%K A143625 cons,easy,nonn
%O A143625 1,1
%A A143625 Peter Bala (pbala(AT)toucansurf.com), Aug 30 2008
%E A143625 Offset corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 
               05 2009