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A005165 Alternating factorials: n! - (n-1)! + (n-2)! - ... 1!.
(Formerly M3892)
11
0, 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, 1226280710981, 19696509177019, 335990918918981, 6066382786809019, 115578717622022981, 2317323290554617019, 48773618881154822981 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Conjecture: for n > 2, smallest prime divisor of a(n) > n. - Gerald McGarvey, Jun 19 2004. Rebuttal: This is not true - Zivkovic (Math. Comp. 68 (1999), pp. 403-409) has demonstrated that 3612703 divides A_{n} for all n >= 3612702. - Paul Jobling, Oct 18 2004.

REFERENCES

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

R. K. Guy, Unsolved Problems in Number Theory, B43.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zivkovic (Math. Comp. 68 (1999), pp. 403-409).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7

E. Wegrzynowski, Series de factorielles

Eric Weisstein's World of Mathematics, Factorial.

Eric Weisstein's World of Mathematics, Alternating Factorial

Index entries for sequences related to factorial numbers

FORMULA

a(0) = 0, a(n) = n! - a(n-1) for n > 0 also a(n) = n*a(n-2) + (n-1)*a(n-1) for n > 1. Sum(n>=1, Pi^n/a(n) ) ~ 30.00005. - Gerald McGarvey, Jun 19 2004

MAPLE

A005165 := proc(n) local i; add((-1)^(n-i)*i!, i=1..n); end;

MATHEMATICA

nn=25; With[{fctrls=Range[nn]!}, Table[Abs[Total[Times@@@Partition[ Riffle[ Take[ fctrls, n], {1, -1}], 2]]], {n, nn}]] (* Harvey P. Dale, Dec 10 2011 *)

a[0] = 0; a[n_] := n! - a[n - 1]; Array[a, 26, 0] (* Robert G. Wilson v, Aug 06 2012 *)

Rest@ LinearRecurrence[{#1 - 1, #1}, {1, 0}, 23] (* Robert G. Wilson v, Jun 15 2013 *)

PROG

(PARI) a(n)=if(n<0, 0, sum(k=0, n-1, (-1)^k*(n-k)!))

(Python)

a = 0

f = 1

for n in range(1, 33):

    print a,

    f *= n

    a = f - a

# from Alex Ratushnyak, Aug 05 2012

(Haskell)

a005165 n = a005165_list !! n

a005165_list = 0 : zipWith (-) (tail a000142_list) a005165_list

-- Reinhard Zumkeller, Jul 21 2013

CROSSREFS

Cf. A000142, A003422, A071828, A001272.

Sequence in context: A106958 A146144 A162292 * A071828 A158615 A088180

Adjacent sequences:  A005162 A005163 A005164 * A005166 A005167 A005168

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 18 18:34 EDT 2014. Contains 240732 sequences.