The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A002467 The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?). (Formerly M3507 N1423) 70
 0, 1, 1, 4, 15, 76, 455, 3186, 25487, 229384, 2293839, 25232230, 302786759, 3936227868, 55107190151, 826607852266, 13225725636255, 224837335816336, 4047072044694047, 76894368849186894, 1537887376983737879, 32295634916658495460, 710503968166486900119 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(n) is the number of permutations in the symmetric group S_n that have a fixed point, i.e., they are not derangements (A000166). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 a(n+1)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=k! for k=0,1,...,n. - Michael Somos, Oct 07 2003 The termwise sum of this sequence and A000166 gives the factorial numbers. - D. G. Rogers, Aug 26 2006, Jan 06 2008 a(n) is the number of deco polyominoes of height n and having in the last column an odd number of cells. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=1 because the horizontal domino is the only deco polyomino of height 2 having an odd number of cells in the last column. - Emeric Deutsch, May 08 2008 Starting (1, 4, 15, 76, 455, ...) = eigensequence of triangle A127899 (unsigned). - Gary W. Adamson, Dec 29 2008 (n-1) | a(n), hence a(n) is never prime. - Jonathan Vos Post, Mar 25 2009 a(n) is the number of permutations of [n] that have at least one fixed point = number of positive terms in n-th row of the triangle in A170942, n > 0. - Reinhard Zumkeller, Mar 29 2012 Numerator of partial sum of alternating harmonic series, provided that the denominator is n!. - Richard Locke Peterson, May 11 2020 REFERENCES R. K. Guy, Unsolved Problems Number Theory, E37. R. K. Guy and R. J. Nowakowski, "Mousetrap," in D. Miklos, V. T. Sos and T. Szonyi, eds., Combinatorics, Paul ErdÅ‘s is Eighty. Bolyai Society Math. Studies, Vol. 1, pp. 193-206, 1993. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe) E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42. P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29. Sergi Elizalde, Bijections for restricted inversion sequences and permutations with fixed points, arXiv:2006.13842 [math.CO], 2020. R. K. Guy, Letter to N. J. A. Sloane, Feb 10 1993 R. K. Guy and R. J. Nowakowski, Mousetrap, Preprint, Feb 10 1993 [Annotated scanned copy] R. K. Guy and S. Washburn, Correspondence, Nov. - Dec. 1991 T. Kotek, J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020 [math.CO], 2013. J. Metzger, Email to N. J. A. Sloane, Apr 30 1991 Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap', European J. Combin. 15 (1994), no. 6, 555-560. 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. Simon Plouffe, Exact formulas for Integer Sequences. A. Steen, Some formulas respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241. L. Takacs, The Problem of Coincidences, Archive for History of Exact Sciences, Volume 21, No. 3, Sept. 1980. pp 229-244, paragraphs 5 and 7. Eric Weisstein's World of Mathematics, Mousetrap FORMULA a(n) = n! - A000166(n) = A000142(n) - A000166(n). E.g.f.: (1 - exp(-x)) / (1 - x). - Michael Somos, Aug 11 1999 a(n) = (n-1)*(a(n-1) + a(n-2)), n > 1; a(1) = 1. - Michael Somos, Aug 11 1999 a(n) = n*a(n-1) - (-1)^n. - Michael Somos, Aug 11 1999 a(0) = 0, a(n) = floor(n!(e-1)/e + 1/2) for n > 0. - Michael Somos, Aug 11 1999 a(n) = - n! * Sum_{i=1..n} (-1)^i/i!. Limit_{n->infinity} a(n)/n! = 1 - 1/e. - Gerald McGarvey, Jun 08 2004 Inverse binomial transform of A002627. - Ross La Haye, Sep 21 2004 a(n) = (n-1)*(a(n-1) + a(n-2)), n > 1. - Gary Detlefs, Apr 11 2010 a(n) = n! - floor((n!+1)/e), n > 0. - Gary Detlefs, Apr 11 2010 For n > 0, a(n) = {(1-1/exp(1))*n!}, where {x} is the nearest integer. - Simon Plouffe, conjectured March 1993, added Feb 17 2011 0 = a(n) * (a(n+1) + a(n+2) - a(n+3)) + a(n+1) * (a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2) * (a(n+2)) if n >= 0. - Michael Somos, Jan 25 2014 a(n) = Gamma(n+1) - Gamma(n+1, -1)*exp(-1). - Peter Luschny, Feb 28 2017 a(n) = Sum_{k=0..n-1} A047920(n-1,k). - Alois P. Heinz, Sep 01 2021 EXAMPLE G.f. = x + x^2 + 4*x^3 + 15*x^4 + 76*x^5 + 455*x^6 + 3186*x^7 + 25487*x^8 + ... MAPLE a := proc(n) -add((-1)^i*binomial(n, i)*(n-i)!, i=1..n) end; a := n->-n!*add((-1)^k/k!, k=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, May 25 2007 a := n -> simplify(GAMMA(n+1) - GAMMA(n+1, -1)*exp(-1)): seq(a(n), n=0..20); # Peter Luschny, Feb 28 2017 MATHEMATICA Denominator[k=1; NestList[1+1/(k++ #1)&, 1, 12]] (* Wouter Meeussen, Mar 24 2007 *) a[ n_] := If[ n < 0, 0, n! - Subfactorial[n]] (* Michael Somos, Jan 25 2014 *) a[ n_] := If[ n < 1, 0, n! - Round[ n! / E]] (* Michael Somos, Jan 25 2014 *) a[ n_] := If[ n < 0, 0, n! - (-1)^n HypergeometricPFQ[ {- n, 1}, {}, 1]](* Michael Somos, Jan 25 2014 *) a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 - Exp[ -x] ) / (1 - x), {x, 0, n}]] (* Michael Somos, Jan 25 2014 *) RecurrenceTable[{a[n] == (n - 1) ( a[n - 1] + a[n - 2]), a[0] == 0, a[1] == 1}, a[n], {n, 20}] (* Ray Chandler, Jul 30 2015 *) PROG (PARI) {a(n) = if( n<1, 0, n * a(n-1) - (-1)^n)} /* Michael Somos, Mar 24 2003 */ (PARI) {a(n) = if( n<0, 0, n! * polcoeff( (1 - exp( -x + x * O(x^n))) / (1 - x), n))} /* Michael Somos, Mar 24 2003 */ (PARI) a(n) = if(n<1, 0, subst(polinterpolate(vector(n, k, (k-1)!)), x, n+1)) (PARI) A002467(n) = if(n<1, 0, n*A002467(n-1)-(-1)^n); \\ Joerg Arndt, Apr 22 2013 CROSSREFS Cf. A002468, A002469, A028306, A047920, A052169, A127899, A276975. Row sums of A068106. Column k=1 of A293211. Column k=0 of A299789, A306234, and of A324362. Sequence in context: A263004 A002750 A178887 * A332652 A243327 A179511 Adjacent sequences:  A002464 A002465 A002466 * A002468 A002469 A002470 KEYWORD nonn,easy,nice AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 3 16:27 EDT 2022. Contains 355055 sequences. (Running on oeis4.)