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A168604
a(n) = 2^(n-2) - 1.
12
1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295, 8589934591
OFFSET
3,2
COMMENTS
Number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly two nonempty parts.
An elephant sequence, seeA175655.For the central square six A[5] vectors, with decimal values between 26 and 176, lead to this sequence. For the corner squares these vectors lead to the companion sequenceA000325(without the first leading 1). -Johannes W. Meijer,Aug 15 2010
FORMULA
E.g.f.: 2*exp(2*x)-exp(x).
a(n) =A000225(n-2).
G.f.: x^3/((1-x)*(1-2*x))
a(n) =A126646(n-3). -R. J. Mathar,Dec 11 2009
a(n) = 3*a(n-1) - 2*a(n-2). -Arkadiusz Wesolowski,Jun 14 2013
a(n) =A000918(n-2) + 1. -Miquel Cerda,Aug 09 2016
EXAMPLE
The partitions of {1,1,1,2,3} into exactly two nonempty parts are {{1},{1,1,2,3}}, {{2},{1,1,1,3}}, {{3},{1,1,1,2}}, {{1,1},{1,2,3}}, {{1,2},{1,1,3}}, {{1,3},{1,1,2}} and {{2,3},{1,1,1}}.
MATHEMATICA
f4[n_]:= 2^(n - 2) - 1; Table[f4[n], {n, 3, 30}]
LinearRecurrence[{3, -2}, {1, 3}, 40] (*Harvey P. Dale,Oct 20 2013 *)
PROG
(Magma) [2^(n-2)-1: n in [3..35]]; //Vincenzo Librandi,May 13 2011
(PARI) a(n)=2^(n-2)-1 \\Charles R Greathouse IV,Oct 07 2015
CROSSREFS
The number of ways of partitioning the multiset {1, 1, 1, 2, 3,..., n-1} into exactly three and four nonempty parts are given inA168605andA168606,respectively.
KEYWORD
nonn,easy
AUTHOR
Martin Griffiths,Dec 01 2009
STATUS
approved