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! + + + - - - ... .

%I A143626 #16 Oct 14 2023 13:25:36
%S A143626 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,
%T A143626 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,
%U A143626 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
%N 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! + + + - - - ...  .
%C A143626 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.
%C A143626 The decimal expansions of E_3(0) and E_3(2) are given in A143635 and A143627. Compare with A143623 and A143624.
%C A143626 E_3(n) as linear combination of E_3(i), i = 0..2.
%C A143626 =======================================
%C A143626 ..E_3(n)..|....E_3(0)...E_3(1)...E_3(2)
%C A143626 =======================================
%C A143626 ..E_3(3)..|.....-1.......-2........3...
%C A143626 ..E_3(4)..|.....-6.......-7........7...
%C A143626 ..E_3(5)..|....-25......-23.......14...
%C A143626 ..E_3(6)..|....-89......-80.......16...
%C A143626 ..E_3(7)..|...-280.....-271......-77...
%C A143626 ..E_3(8)..|...-700.....-750.....-922...
%C A143626 ..E_3(9)..|...-380.....-647....-6660...
%C A143626 ..E_3(10).|..13452....13039...-41264...
%C A143626 ...
%C A143626 The columns are A143628, A143629 and A143630.
%e A143626 1.3015594959829796028430427
%t A143626 RealDigits[ N[ (4*E^(3/2)*Cos[Sqrt[3]/2] - 1)/(3*E), 105]][[1]] (* _Jean-François Alcover_, Nov 08 2012 *)
%Y A143626 Cf. A143623, A143624, A143625, A143627, A143628, A143629, A143630.
%K A143626 cons,easy,nonn
%O A143626 1,2
%A A143626 _Peter Bala_, Aug 30 2008
%E A143626 Offset corrected by _R. J. Mathar_, Feb 05 2009