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)

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

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

T. D. Noe, Table of n, a(n) for n = 0..200

Eric Weisstein's World of Mathematics, Multifactorial.

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

a(3*n) = A032031(n); a(3*n+1) = A007559(n+1); a(3*n+2) = A008544(n+1). - Reinhard Zumkeller, Sep 20 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

Sum_{n>=0} 1/a(n) = A288055. - Amiram Eldar, Nov 10 2020

Maple

A007661 := n -> mul(k, k = select(k -> k mod 3 = n mod 3, [$1 .. n])): seq(A007661(n), n = 0 .. 29);  # Peter Luschny, Jun 23 2011

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

(PARI) A007661(n, d=3)=prod(i=0, (n-1)\d, n-d*i) \\ M. F. Hasler, Feb 16 2008
(Haskell)
a007661 n k = a007661_list !! n
a007661_list = 1 : 1 : 2 : zipWith (*) a007661_list [3..]
-- Reinhard Zumkeller, Sep 20 2013
(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)
[a(n) for n in (0..30)] # G. C. Greubel, Aug 21 2019
(GAP)
a:= function(n)
    if n<3 then return Fibonacci(n+1);
    else return n*a(n-3);
    fi;
  end;
List([0..30], n-> a(n) ); # G. C. Greubel, Aug 21 2019

Crossrefs

Union of A007559, A008544 and A032031.

Cf. A000142, A006882 (= A001147 union A000165), A007662 (= union of A007696, A001813, A008545 and A047053), A085157, A085158.

Cf. A008585, A016777, A016789, A161474, A288055.

Sequence in context: A329660 A098088 A080500 * A049891 A135432 A108364

Adjacent sequences: A007658 A007659 A007660 * A007662 A007663 A007664

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Status

approved