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%I #53 Sep 20 2022 10:16:53
%S 1,2,4,9,32,150,864,5880,46080,408240,4032000,43908480,522547200,
%T 6745939200,93884313600,1401079680000,22317642547200,377917892352000,
%U 6778983923712000,128403161542656000,2560949482291200000,53645489280294912000,1177524571957493760000,27027108408834293760000
%N a(n) = n! + (n-1)! + (n-2)!.
%C In factorial base representation (A007623) the terms are written as: 1, 10, 20, 111, 1110, 11100, 111000, ... From a(3) = 9 = "111" onward each term begins always with three consecutive 1's, followed by n-3 zeros. - _Antti Karttunen_, Sep 24 2016
%H Vincenzo Librandi, <a href="/A054119/b054119.txt">Table of n, a(n) for n = 0..300</a>
%H Michele Battagliola, Nadir Murru, and Giordano Santilli, <a href="https://arxiv.org/abs/2209.08875">Combinatorial properties of multidimensional continued fractions</a>, arXiv:2209.08875 [math.NT], 2022. See Proposition 3.3 pp. 6-7.
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F For n>2, a(n) = (n-2)! * n^2. [_Gary Detlefs_, Aug 01 2009]
%F a(n) = (n+1)!*(H(n-1)+H(n+1)-H(n-2)-H(n))/2, n>1, where H(n) is the n-th harmonic number. [_Gary Detlefs_, Oct 04 2011]
%F E.g.f.: x + 1/(1-x) - x*log(1-x) = x^2/G(0)/2 where G(k) = 1 + (k+2)/(x - x*(k+1)/(x + k + 1 - x^4/(x^3 +(k+2)*(k+3)/G(k+1)))); (continued fraction, 3rd kind, 4-step). - _Sergei N. Gladkovskii_, Jul 06 2012
%F G.f.: G(0) where G(k) = 1 - x/(1 + x/(1 - x - (k+1)/( k+1 - x/Q))); (continued fraction, 3rd kind, 4-step). - _Sergei N. Gladkovskii_, Jul 28 2012
%F For n >= 1, a(n) = A276940(n)/n. - _Antti Karttunen_, Sep 24 2016
%F Sum_{n>=2} 1/a(n) = A306770. - _Amiram Eldar_, Nov 19 2020
%p f:= n-> `if`(n<0, 0, n!):
%p seq(f(n)+f(n-1)+f(n-2), n=0..23);
%t Join[{1,2},Table[n!+(n+1)!+(n+2)!,{n,0,30}]] (* _Vladimir Joseph Stephan Orlovsky_, May 19 2011 *)
%o (Magma) [1,2],[Factorial(n)+Factorial(n-1)+Factorial(n-2): n in [2..20]]; // _Vincenzo Librandi_, Oct 05 2011
%o (Scheme) (define (A054119 n) (if (<= n 1) (+ 1 n) (+ (A000142 n) (A000142 (- n 1)) (A000142 (- n 2))))) ;; _Antti Karttunen_, Sep 24 2016
%o (PARI) f(n) = if (n<0, 0, n!);
%o a(n) = f(n) + f(n-1) + f(n-2); \\ _Michel Marcus_, Sep 20 2022
%Y Cf. A001048, A030495, A108217, A276940, A306770.
%Y Equals T(n, 3), array T as in A054115.
%Y Row 6 of A276955 (from a(3)=9 onward).
%K nonn
%O 0,2
%A _Clark Kimberling_
%E Simpler definition from _Miklos Kristof_, Jun 16 2005
%E More terms from _Antti Karttunen_, Sep 24 2016
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