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%I M3962 N1635 #59 Aug 04 2022 05:09:41
%S 0,0,1,5,31,203,1501,12449,114955,1171799,13082617,158860349,
%T 2085208951,29427878435,444413828821,7151855533913,122190894996451,
%U 2209057440250799,42133729714051825,845553296311189109,17810791160738752207,392911423093684031099
%N The game of Mousetrap with n cards: the number of permutations of n cards in which 2 is the only hit.
%D R. K. Guy, Unsolved Problems Number Theory, E37.
%D R. K. Guy and R. J. Nowakowski, "Mousetrap," in D. Miklos, V. T. Sos and T. Szonyi, eds., Combinatorics, Paul Erdős is Eighty. Bolyai Society Math. Studies, Vol. 1, pp. 193-206, 1993.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002469/b002469.txt">Table of n, a(n) for n = 2..100</a>
%H R. K. Guy and R. J. Nowakowski, <a href="/A002467/a002467_1.pdf">Mousetrap</a>, Preprint, Feb 10 1993 [Annotated scanned copy]
%H J. Metzger, <a href="/A002467/a002467_3.pdf">Email to N. J. A. Sloane, Apr 30 1991</a>
%H Daniel J. Mundfrom, <a href="http://dx.doi.org/10.1006/eujc.1994.1057">A problem in permutations: the game of 'Mousetrap'</a>. European J. Combin. 15 (1994), no. 6, 555-560.
%H M. Z. Spivey, <a href="http://dx.doi.org/10.1016/j.ejc.2008.04.005">Staircase rook polynomials and Cayley's game of mousetrap</a>, Eur. J. Combinat. 30 (2) (2009) 532-539
%H A. Steen, <a href="http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0015/DMDLOG_0031">Some formulas respecting the game of mousetrap</a>, Quart. J. Pure Applied Math., 15 (1878), 230-241.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Mousetrap.html">Mousetrap</a>
%F a(n) = sum of terms in (n-2)-nd row of triangle A159610; equivalent to: a(n) = (n-2)*A000255(n-1) + A000166(n). - _Gary W. Adamson_, Apr 17 2009
%F a(n) = (n-3)* A000166(n-2) + (n-4)* A000166(n-3). - _Gary Detlefs_, Apr 10 2010
%F a(n) = (n-3)*floor(((n-2)!+1)/e) + (n-4)*floor(((n-3)!+1)/e), for n>2. - _Gary Detlefs_, Apr 10 2010
%F G.f.: x - 1 + (1-2*x)/(x*Q(0)), where Q(k) = 1/x - (2*k+1) - (k+1)*(k+2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Apr 25 2013
%e G.f.: x^4 + 5*x^5 + 31*x^6 + 203*x^7 + 1501*x^8 + 12449*x^9 + 114955*x^10 + ...
%p A002469:=n->(n-3)*floor(((n-2)!+1)/exp(1)) + (n-4)*floor(((n-3)!+1)/exp(1)): 0, seq(A002469(n), n=3..30); # _Wesley Ivan Hurt_, Jan 10 2017
%t Join[{0},Table[(n-3)Floor[((n-2)!+1)/E]+(n-4)Floor[((n-3)!+1)/E], {n,3,30}]] (* _Harvey P. Dale_, Feb 05 2012 *)
%t a[n_] := (n-3)*Subfactorial[n-2]+(n-4)*Subfactorial[n-3]; a[n_ /; n <= 3] = 0; Table[a[n], {n, 2, 23}] (* _Jean-François Alcover_, Dec 12 2014 *)
%o (PARI)
%o default(realprecision,200);
%o e=exp(1);
%o A002469(n) = if( n<=3, 0, (n-3)*floor(((n-2)!+1)/e) + (n-4)*floor(((n-3)!+1)/e) );
%o /* _Joerg Arndt_, Apr 22 2013 */
%Y Cf. A002468, A002467, A028306, A159610, A000255, A000166.
%K nonn,nice
%O 2,4
%A _N. J. A. Sloane_
%E More terms from _Harvey P. Dale_, Feb 05 2012
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