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%I #16 Jun 10 2020 05:12:16
%S -3,-21,-117,-717,-5037,-40317,-362877,-3628797,-39916797,-479001597,
%T -6227020797,-87178291197,-1307674367997,-20922789887997,
%U -355687428095997,-6402373705727997,-121645100408831997,-2432902008176639997
%N a(n) = 1 - Sum_{k=2..n} k*k!.
%D Appeared in University of Texas Interscholastic League High School Number Sense District Test, 2004.
%H G. C. Greubel, <a href="/A092634/b092634.txt">Table of n, a(n) for n = 2..445</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F Conjecture: a(n) +(-n-3)*a(n-1) +(2*n+1)*a(n-2) +(-n+1)*a(n-3)=0. - _R. J. Mathar_, Sep 27 2014
%F a(n) = 3 - (n+1)!. - _Michel Marcus_, Jun 07 2020
%F From _G. C. Greubel_, Jun 07 2020: (Start)
%F a(n) = 3 - !(n+2) + !(n+1) = 3 - A003422(n+2) + A003422(n+1).
%F E.g.f.: 3*exp(x) - (3 - 3*x + x^3)/(1-x)^2. (End)
%p A092634:= n-> 3-(n+1)!: seq(A092634(n), n=2..20); # _G. C. Greubel_, Jun 07 2020
%t Table[i; 1 - Sum[n n!, {n, 2, i}], {i, 2, 20}]
%o (PARI) a(n) = 1 - sum(k=2, n, k*k!); \\ _Michel Marcus_, Jun 07 2020
%o (Sage) [3-factorial(n+1) for n in (2..20)] # _G. C. Greubel_, Jun 07 2020
%Y Cf. A033312.
%K sign
%O 2,1
%A Doug Ray (draymath(AT)iastate.edu), Apr 11 2004
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