OFFSET
0,2
COMMENTS
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
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.
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
MAPLE
f:= n-> `if`(n<0, 0, n!):
seq(f(n)+f(n-1)+f(n-2), n=0..23);
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 *)
PROG
(Magma) [1, 2], [Factorial(n)+Factorial(n-1)+Factorial(n-2): n in [2..20]]; // Vincenzo Librandi, Oct 05 2011
(Scheme) (define (A054119 n) (if (<= n 1) (+ 1 n) (+ (A000142 n) (A000142 (- n 1)) (A000142 (- n 2))))) ;; Antti Karttunen, Sep 24 2016
(PARI) f(n) = if (n<0, 0, n!);
a(n) = f(n) + f(n-1) + f(n-2); \\ Michel Marcus, Sep 20 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Simpler definition from Miklos Kristof, Jun 16 2005
More terms from Antti Karttunen, Sep 24 2016
STATUS
approved