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 A007489 a(n) = Sum_{k=1..n} k!. (Formerly M2818) 90
 0, 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913, 43954713, 522956313, 6749977113, 93928268313, 1401602636313, 22324392524313, 378011820620313, 6780385526348313, 128425485935180313, 2561327494111820313, 53652269665821260313 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Equals row sums of triangle A143122 starting (1, 3, 9, 33, ...). - Gary W. Adamson, Jul 26 2008 A007489(n) for n>=4 is never a perfect square. - Alexander R. Povolotsky, Oct 16 2008 Number of cycles that can be written in the form (j,j+1,j+2,...), in all permutations of {1,2,...,n}. Example: a(3)=9 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), (132) we have 3+2+2+1+1+0=9 such cycles. - Emeric Deutsch, Jul 14 2009 Conjectured to be the length of the shortest word over {1,...,n} that contains each of the n! permutations as a factor (cf. A180632) [see Johnston]. - N. J. A. Sloane, May 25 2013 The above conjecture has been disproven for n>=6. See A180632 and the Houston 2014 reference. - Dmitry Kamenetsky, Mar 07 2016 a(n) is also the number of compositions of n if cardinal values do not matter but ordinal rankings do. Since cardinal values do not matter, a sequence of k summands summing to n can be represented as (s(1),...,s(k)), where the s's are positive integers and the numbers in parentheses are the initial ordinal rankings. The number of compositions of these summands are equal to k!, with k ranging from 1 to n. - Gregory L. Simay, Jul 31 2016 When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the left. Compare array A211370 for circular shifts to the left in a broader sense. Compare sequence A001563 for circular shifts to the right. - Tilman Piesk, Apr 29 2017 Since a(n) = (1!+2!+3!+...+n!) = 3(1+3!/3+4!/3+...+n!/3) is a multiple of 3 for n>2, the only prime in this sequence is a(2) = 3. - Eric W. Weisstein, Jul 15 2017 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Carauleanu Marc and T. D. Noe, Table of n, a(n) for n = 0..212 (first 100 terms from T. D. Noe) R. K. Guy, Letter to N. J. A. Sloane, 1987 Robin Houston, Tackling the Minimal Superpermutation Problem, arXiv:1408.5108 [math.CO], 2014. Nathaniel Johnston, The minimal superpermutation problem (2013) Nathaniel Johnston, Non-uniqueness of minimal superpermutations, Discrete Math. 313 (2013), no. 14, 1553--1557. MR3047396 S. Legendre, P. Paclet, On the Permutations Generated by Cyclic Shift , J. Int. Seq. 14 (2011) # 11.3.2 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 Eric Weisstein's World of Mathematics, Factorial Eric Weisstein's World of Mathematics, Left Factorial G. Xiao, Sigma Server, Operate on "n!" FORMULA a(n) = Sum_{k=1..n} P(n, k)/C(n, k). - Ross La Haye, Sep 21 2004 a(n) = 3*A056199(n) for n>=2. - Philippe Deléham, Feb 10 2007 a(n) = !(n+1)+1=A003422(n+1)+1. - Artur Jasinski, Nov 08 2007 Starting (1, 3, 9, 33, 153, ...), = row sums of triangle A137593 - Gary W. Adamson, Jan 28 2008 a(n) = a(n-1) + n! for n >= 1. - Jaroslav Krizek, Jun 16 2009 E.g.f. A(x) satisfies to the differential equation A'(x)=A(x)+x/(1-x)^2+1. - Vladimir Kruchinin, Jan 22 2011 a(0)=0, a(1)=1, a(n) = (n+1)*a(n-1)-n*a(n-2). - Sergei N. Gladkovskii, Jul 05 2012 G.f.: W(0)*x/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013 G.f.: x /(1-x)/Q(0),m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013 E.g.f.: exp(x-1)*(Ei(1) - Ei(1-x)) - exp(x) + 1/(1 - x), where Ei(x) is the exponential integral. - Ilya Gutkovskiy, Nov 27 2016 EXAMPLE a(4) = 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33. - Michael B. Porter, Aug 03 2016 MAPLE A007489 := proc(n) local i; add(i!, i=1..n); end; MATHEMATICA FoldList[Plus, 0, (Range@ 21)! ] (* Robert G. Wilson v, Sep 21 2007 *) Table[Sum[i!, {i, 1, n}], {n, 0, 21}] (* Zerinvary Lajos, Jul 12 2009 *) Accumulate[Range!]  (* Harvey P. Dale, Apr 30 2011 *) Table[Plus@@(Range[n]!), {n, 20}] (* Alonso del Arte, Jul 18 2011 *) PROG (PARI) a(n)=sum(k=1, n, k!) \\ Charles R Greathouse IV, Jul 25 2011 (Haskell) a007489 n = a007489_list !! n a007489_list = scanl (+) 0 \$ tail a000142_list -- Reinhard Zumkeller, Aug 29 2014 (MAGMA)  cat [&+[Factorial(i): i in [1..n]]: n in [1..25]]; // Vincenzo Librandi, Sep 02 2016 (GAP) List([1..20], n->Sum([1..n], Factorial)); # Muniru A Asiru, Jan 31 2018 CROSSREFS Equals A003422(n+1) - 1. Cf. A000142, A000670, A137593, A143122, A161128, A180632. Column k=0 of A120695. Sequence in context: A279840 A009220 A294035 * A294638 A201968 A264237 Adjacent sequences:  A007486 A007487 A007488 * A007490 A007491 A007492 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified May 21 22:24 EDT 2019. Contains 323467 sequences. (Running on oeis4.)