A168604 a(n) = 2^(n-2) - 1.

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

Author

Martin Griffiths, Dec 01 2009

Keywords

• nonn
• easy

Comments

Number of ways of partitioning the multiset {1,1,1,2,3,...,n-2} into exactly two nonempty parts.

An elephant sequence, see A175655. 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 sequence A000325 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010

Examples

			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}}.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..1000
• M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

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 in A168605 and A168606, respectively.

Programs

Magma

[2^(n-2)-1 : n in [3..35]]; // Vincenzo Librandi, May 13 2011

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 *)

PARI

a(n)=2^(n-2)-1 \\ Charles R Greathouse IV, Oct 07 2015

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