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A024012 a(n) = 2^n - n^2. 24
1, 1, 0, -1, 0, 7, 28, 79, 192, 431, 924, 1927, 3952, 8023, 16188, 32543, 65280, 130783, 261820, 523927, 1048176, 2096711, 4193820, 8388079, 16776640, 33553807, 67108188, 134216999, 268434672, 536870071, 1073740924, 2147482687, 4294966272, 8589933503, 17179868028, 34359737143 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
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
REFERENCES
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 92.
LINKS
Anant Godbole and Martha Liendo, Waiting time distribution for the emergence of superpatterns, arxiv 1302.4668 [math.PR], 2013.
FORMULA
G.f.: (1 - 4*x + 4*x^2 + x^3)/((1 - 2*x)*(1 - x)^3). - Vincenzo Librandi, Jul 13 2012
a(n) = 5*a(n - 1) - 9*a(n - 2) + 7*a(n - 3) - 2*a(n - 4). - Vincenzo Librandi, Jul 13 2012
MAPLE
seq(2^n-n^2, n=0..35); # Zerinvary Lajos, Jul 01 2007
MATHEMATICA
CoefficientList[Series[(1 - 4*x + 4*x^2 + x^3)/((1 - x)^3(1 - 2x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 13 2012 *)
Table[2^n - n^2, {n, 0, 39}] (* Alonso del Arte, Dec 16 2012 *)
PROG
(Magma) [2^n-n^2: n in [0..30]]; // Vincenzo Librandi, Apr 29 2011
(PARI) a(n)=2^n-n^2 \\ Charles R Greathouse IV, Apr 17 2012
(Maxima) A024012(n):=2^n-n^2$ makelist(A024012(n), n, 0, 20); /* Martin Ettl, Dec 18 2012 */
CROSSREFS
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).
Sequence in context: A341067 A061968 A223772 * A352900 A163705 A162595
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
More terms from Hugo Pfoertner, Oct 18 2004
STATUS
approved

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Last modified February 28 14:45 EST 2024. Contains 370400 sequences. (Running on oeis4.)