|
|
A007661
|
|
Triple factorial numbers a(n) = n!!!, defined by a(n) = n*a(n-3), a(0) = a(1) = 1, a(2) = 2. Sometimes written n!3.
(Formerly M0596)
|
|
118
|
|
|
1, 1, 2, 3, 4, 10, 18, 28, 80, 162, 280, 880, 1944, 3640, 12320, 29160, 58240, 209440, 524880, 1106560, 4188800, 11022480, 24344320, 96342400, 264539520, 608608000, 2504902400, 7142567040, 17041024000, 72642169600, 214277011200, 528271744000, 2324549427200
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
The triple factorial of a positive integer n is the product of the positive integers <= n that have the same residue modulo 3 as n. - Peter Luschny, Jun 23 2011
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = Product_{i=0..floor((n-1)/3)} (n-3*i). - M. F. Hasler, Feb 16 2008
a(n) ~ c * n^(n/3+1/2)/exp(n/3), where c = sqrt(2*Pi/3) if n=3*k, c = sqrt(2*Pi)*3^(1/6) / Gamma(1/3) if n=3*k+1, c = sqrt(2*Pi)*3^(-1/6) / Gamma(2/3) if n=3*k+2. - Vaclav Kotesovec, Jul 29 2013
0 = a(n)*(a(n+1) -a(n+4)) +a(n+1)*a(n+3) for all n>=0. - Michael Somos, Feb 24 2019
|
|
MAPLE
|
|
|
MATHEMATICA
|
multiFactorial[n_, k_] := If[n < 1, 1, If[n < k + 1, n, n*multiFactorial[n - k, k]]]; Array[ multiFactorial[#, 3] &, 30, 0] (* Robert G. Wilson v, Apr 23 2011 *)
RecurrenceTable[{a[0]==a[1]==1, a[2]==2, a[n]==n*a[n-3]}, a, {n, 30}] (* Harvey P. Dale, May 17 2012 *)
Table[With[{q = Quotient[n + 2, 3]}, 3^q q! Binomial[n/3, q]], {n, 0, 30}] (* Jan Mangaldan, Mar 21 2013 *)
a[ n_] := With[{m = Mod[n, 3, 1], q = 1 + Quotient[n, 3, 1]}, If[n < 0, 0, 3^q Pochhammer[m/3, q]]]; (* Michael Somos, Feb 24 2019 *)
Table[Times@@Range[n, 1, -3], {n, 0, 30}] (* Harvey P. Dale, Sep 12 2020 *)
|
|
PROG
|
(Haskell)
a007661 n k = a007661_list !! n
a007661_list = 1 : 1 : 2 : zipWith (*) a007661_list [3..]
(Magma) I:=[1, 1, 2]; [n le 3 select I[n] else (n-1)*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Nov 27 2015
(Sage)
def a(n):
if (n<3): return fibonacci(n+1)
else: return n*a(n-3)
(GAP)
a:= function(n)
if n<3 then return Fibonacci(n+1);
else return n*a(n-3);
fi;
end;
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|