%I #24 Jun 13 2015 00:50:48

%S 0,0,1,1,0,1,3,2,0,2,5,3,0,3,7,4,0,4,9,5,0,5,11,6,0,6,13,7,0,7,15,8,0,

%T 8,17,9,0,9,19,10,0,10,21,11,0,11,23,12,0,12,25,13,0,13,27,14,0,14,29,

%U 15,0,15,31,16,0,16,33,17,0,17,35,18,0,18,37,19,0,19,39,20,0,20,41,21,0

%N a(n) = #{0<=k<=n: mod(kn,4)=2}.

%C Coefficients in the unique expansion of e/4 = Sum[a(n)/n!, n>=1], where a(n) satisfies 0<=a(n)<n-1.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,4,-3,2,-1).

%F a(n)=0 if n=4k, a(n)=k if n=4k+1, a(n)=2k+1 if n=4k+2 and a(n)=k+1 if n=4k+3.

%F a(n) = floor(n!*e/4) - n*floor((n-1)!*e/4). - _Benoit Cloitre_, Dec 07 2002

%F a(n) = Sum_{k=0..n} if (mod(nk, 4)=2, 1, 0)}. E.g. a(6) = #{1, 3, 5} = 3. - _Paul Barry_, Sep 10 2003

%F O.g.f.: x^2*(1-x+x^2)/((x-1)^2*(1+x^2)^2). - _R. J. Mathar_, Jun 13 2008

%F From _Wesley Ivan Hurt_, May 30 2015: (Start)

%F a(n) = 2*a(n-1)-3*a(n-2)+4*a(n-3)-3*a(n-4)+2*a(n-5)-a(n-6), n>6.

%F a(n) = (-1)^((1-2*n-(-1)^n)/4)*((-1)^n-2*n*(-1)^((2*n+3+(-1)^n)/4)+n*(-1)^((1+(-1)^n)/2)+n*(-1)^((2*n+1+(-1)^n)/2)-1)/8. (End)

%e sum(i=1,10,a(i)/i!)=0.6795703813... exp(1)/4=0.679570457...

%t a = Table[0, {i, 1, 50}]; x = Exp[1]/4; For[n = 2, n <= 51, n++, { an = 0; While [(x >= (1/n!)) && (an < (n - 1)), {an++, x = x - (1/n!)} ]}; a[[n - 1]] = an;]; a

%Y Cf. A009947, A087509, A087620.

%K nonn,easy

%O 0,7

%A _John W. Layman_, Dec 03 2002

%E More terms from _Benoit Cloitre_, Dec 07 2002