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%I #45 Jan 06 2025 08:04:06
%S 1,1,0,-1,0,7,28,79,192,431,924,1927,3952,8023,16188,32543,65280,
%T 130783,261820,523927,1048176,2096711,4193820,8388079,16776640,
%U 33553807,67108188,134216999,268434672,536870071,1073740924,2147482687,4294966272,8589933503,17179868028,34359737143
%N a(n) = 2^n - n^2.
%C The sequence 2^(n-2) - (n-2)^2, n=7,8,... enumerates the number of non-isomorphic sequences of length n, with entries from {1,2,3} and no two adjacent entries the same, that contain each of the thirteen rankings of three players (111, 121, 112, 211, 122, 212, 221, 123, 132, 213, 231, 312, 321) as embedded order isomorphic subsequences. See the arXiv paper below for proof. If n=7, these sequences are 1213121, 1213212, 1231213, 1231231,1231321, 1232123, and 1232132, and for each case, there are 3!=6 isomorphs. - _Anant Godbole_, Feb 20 2013
%D GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 92.
%H Vincenzo Librandi, <a href="/A024012/b024012.txt">Table of n, a(n) for n = 0..1000</a>
%H Anant Godbole and Martha Liendo, <a href="http://arxiv.org/abs/1302.4668">Waiting time distribution for the emergence of superpatterns</a>, arxiv 1302.4668 [math.PR], 2013.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).
%F G.f.: (1 - 4*x + 4*x^2 + x^3)/((1 - 2*x)*(1 - x)^3). - _Vincenzo Librandi_, Jul 13 2012
%F a(n) = 5*a(n - 1) - 9*a(n - 2) + 7*a(n - 3) - 2*a(n - 4). - _Vincenzo Librandi_, Jul 13 2012
%p seq(2^n-n^2, n=0..35); # _Zerinvary Lajos_, Jul 01 2007
%t CoefficientList[Series[(1 - 4*x + 4*x^2 + x^3)/((1 - x)^3(1 - 2x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 13 2012 *)
%t Table[2^n - n^2, {n, 0, 39}] (* _Alonso del Arte_, Dec 16 2012 *)
%o (Magma) [2^n-n^2: n in [0..30]]; // _Vincenzo Librandi_, Apr 29 2011
%o (PARI) a(n)=2^n-n^2 \\ _Charles R Greathouse IV_, Apr 17 2012
%o (Maxima) A024012(n):=2^n-n^2$ makelist(A024012(n),n,0,20); /* _Martin Ettl_, Dec 18 2012 */
%Y Cf. A072180 (2^n - n^2 is prime), A075896 (primes of the form 2^n - n^2), A099481 (2^n - n^2 is a semiprime), A099482 (semiprimes of the form 2^n - n^2).
%K sign,easy
%O 0,6
%A _N. J. A. Sloane_.
%E More terms from _Hugo Pfoertner_, Oct 18 2004