A143625 - OEIS

%I #20 Oct 14 2023 13:26:01

%S 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,

%T 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,

%U 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

%N 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! + + + - - - ...  .

%C 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 The decimal expansions of E_3(1) and E_3(2) are given in A143626 and A143627. Compare with A143623 and A143624.

%C E_3(n) as linear combination of E_3(i), i = 0..2.

%C =======================================

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

%C =======================================

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

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

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

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

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

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

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

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

%C ...

%C The columns are A143628, A143629 and A143630.

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

%e 2.284942382409635208999050...

%t 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 *)

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

%K cons,easy,nonn

%O 1,1

%A _Peter Bala_, Aug 30 2008

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