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A007661
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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)
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118
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Multifactorial.
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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.
Sequence in context: A055506 A098088 A080500 * A049891 A135432 A108364
Adjacent sequences: A007658 A007659 A007660 * A007662 A007663 A007664
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v
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STATUS
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approved
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