A054119 a(n) = n! + (n-1)! + (n-2)!.
1, 2, 4, 9, 32, 150, 864, 5880, 46080, 408240, 4032000, 43908480, 522547200, 6745939200, 93884313600, 1401079680000, 22317642547200, 377917892352000, 6778983923712000, 128403161542656000, 2560949482291200000, 53645489280294912000, 1177524571957493760000, 27027108408834293760000
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
- Michele Battagliola, Nadir Murru, and Giordano Santilli, Combinatorial properties of multidimensional continued fractions, arXiv:2209.08875 [math.NT], 2022. See Proposition 3.3 pp. 6-7.
- Index entries for sequences related to factorial base representation
Crossrefs
Programs
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Magma
[1,2] cat [Factorial(n)+Factorial(n-1)+Factorial(n-2): n in [2..20]]; // Vincenzo Librandi, Oct 05 2011
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Maple
f:= n-> `if`(n<0, 0, n!): seq(f(n)+f(n-1)+f(n-2), n=0..23);
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Mathematica
Join[{1,2},Table[n!+(n+1)!+(n+2)!,{n,0,30}]] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *) Join[{1,2,4},Plus@@@Partition[Range[30]!,3,1]] (* Harvey P. Dale, Aug 29 2024 *) -
PARI
f(n) = if (n<0, 0, n!); a(n) = f(n) + f(n-1) + f(n-2); \\ Michel Marcus, Sep 20 2022
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Scheme
(define (A054119 n) (if (<= n 1) (+ 1 n) (+ (A000142 n) (A000142 (- n 1)) (A000142 (- n 2))))) ;; Antti Karttunen, Sep 24 2016
Formula
For n>2, a(n) = (n-2)! * n^2. [Gary Detlefs, Aug 01 2009]
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]
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
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
For n >= 1, a(n) = A276940(n)/n. - Antti Karttunen, Sep 24 2016
Sum_{n>=2} 1/a(n) = A306770. - Amiram Eldar, Nov 19 2020
Extensions
Simpler definition from Miklos Kristof, Jun 16 2005
More terms from Antti Karttunen, Sep 24 2016
Comments