A024012 a(n) = 2^n - n^2.

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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Anant Godbole and Martha Liendo, Waiting time distribution for the emergence of superpatterns, arxiv 1302.4668 [math.PR], 2013.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

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

Adjacent sequences: A024009 A024010 A024011 * A024013 A024014 A024015

Keyword

sign,easy

Author

N. J. A. Sloane.

Extensions

More terms from Hugo Pfoertner, Oct 18 2004

Status

approved