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A083329 a(0) = 1; for n > 0, a(n) = 3*2^(n-1) - 1. 41
1, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Apart from leading term (which should really be 3/2), same as A055010.

Binomial transform of A040001. Inverse binomial transform of A053156.

a(n) = A105728(n+1,2). - Reinhard Zumkeller, Apr 18 2005

Row sums of triangle A133567. - Gary W. Adamson, Sep 16 2007

Row sums of triangle A135226. - Gary W. Adamson, Nov 23 2007

a(n) = number of partitions Pi of [n+1] (in standard increasing form) such that the permutation Flatten[Pi] avoids the patterns 2-1-3 and 3-1-2. Example: a(3)=11 counts all 15 partitions of [4] except 13/24, 13/2/4 which contain a 2-1-3 and 14/23, 14/2/3 which contain a 3-1-2. Here "standard increasing form" means the entries are increasing in each block and the blocks are arranged in increasing order of their first entries. - David Callan, Jul 22 2008

An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 42, 138, 162, 168, lead to this sequence. For the central square these vectors lead to the companion sequence A003945. - Johannes W. Meijer, Aug 15 2010

The binary representation of a(n) has n+1 digits, where all digits are 1's except digit n-1. For example: a(4) = 23 = 10111 (2). - Omar E. Pol, Dec 02 2012

Row sums of triangle A209561. - Reinhard Zumkeller, Dec 26 2012

If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/A047679, A162909/A162910, A071766/A229742, A245325/A245326, ...), and if it is organised by blocks or levels (n) with 2^n terms (n>=0), and the fractions, term by term, are summed at each level n, then the resulting sequence of integers is a(n)+1/2, apart from leading term (which should be 1/2). - Yosu Yurramendi, May 23 2015

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5

Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)

Index entries for linear recurrences with constant coefficients, signature (3,-2).

FORMULA

a(n) = (3*2^n - 2 + 0^n)/2.

G.f.: (1-x+x^2)/((1-x)*(1-2*x)).

E.g.f.: (3*exp(2*x)-2*exp(x)+exp(0))/2.

a(0) = 1, a(n) = sum of all previous terms + n. - Amarnath Murthy, Jun 20 2004

a(n) = 3*a(n-1)-2*a(n-2) for n>2, a(0)=1, a(1)=2, a(2)=5. - Philippe Deléham, Nov 29 2013

From Bob Selcoe, Apr 25 2014: (Start)

a(n) = (...((((((1)+1)*2+1)*2+1)*2+1)*2+1)...), with n+1 1's, n >= 0.

a(n) = 2*a(n-1) + 1, n >= 2.

a(n) = 2^n + 2^(n-1) - 1, n >= 2. (End)

EXAMPLE

a(0) = (3*2^0 - 2 + 0^0)/2 = 2/2 = 1 (use 0^0=1).

MAPLE

seq(ceil((2^i+2^(i+1)-2)/2), i=0..31); # Zerinvary Lajos, Oct 02 2007

MATHEMATICA

a[1] = 2; a[n_] := 2a[n - 1] + 1; Table[ a[n], {n, 31}] (* Robert G. Wilson v, May 04 2004 *)

Join[{1}, LinearRecurrence[{3, -2}, {2, 5}, 40]] (* Vincenzo Librandi, Jan 01 2016 *)

PROG

(Haskell)

a083329 n = a083329_list !! n

a083329_list = 1 : iterate ((+ 1) . (* 2)) 2

-- Reinhard Zumkeller, Dec 26 2012, Feb 22 2012

(PARI) a(n)=(3*2^n-2+0^n)/2 \\ Charles R Greathouse IV, Sep 24 2015

(MAGMA) [1] cat [3*2^(n-1)-1: n in [1..40]]; // Vincenzo Librandi, Jan 01 2016

CROSSREFS

Essentially the same as A055010 and A052940.

Cf. A000225, A052955, A133567, A135226.

Cf. A007505 (primes).

Cf. A266550 (independence number of the n-Mycielski graph).

Sequence in context: A086219 A055010 A153893 * A266550 A081973 A055496

Adjacent sequences:  A083326 A083327 A083328 * A083330 A083331 A083332

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Apr 27 2003

EXTENSIONS

The generating function corrected by Martin Griffiths (griffm(AT)essex.ac.uk), Dec 01 2009

STATUS

approved

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Last modified December 3 01:12 EST 2016. Contains 278694 sequences.