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A083329
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a(0) = 1; for n > 0, 3*2^(n-1) - 1.
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28
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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S. Kitaev, J. Remmel and M. Tiefenbruck, Quadrant marked mesh patterns in 132-avoiding permutations II, arXiv preprint arXiv:1302.2274, 2013
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Mycielski Graph - Eric W. Weisstein, Nov 24 2008
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FORMULA
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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 (amarnath_murthy(AT)yahoo.com), Jun 20 2004
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EXAMPLE
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a(0)=(3*2^0-2+0^0)/2=2/2=1 (use 0^0=1).
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MAPLE
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seq(ceil((2^i+2^(i+1)-2)/2), i=0..31); # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007
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MATHEMATICA
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a[1] = 2; a[n_] := 2a[n - 1] + 1; Table[ a[n], {n, 31}] (* Robert G. Wilson v, May 04 2004 *)
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PROG
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(Haskell)
a083329 n = a083329_list !! n
a083329_list = 1 : iterate ((+ 1) . (* 2)) 2
-- Reinhard Zumkeller, Dec 26 2012, Feb 22 2012
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CROSSREFS
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Essentially the same as A055010 and A052940.
Cf. A000225, A052955, A133567, A135226.
Sequence in context: A086219 A055010 * A153893 A081973 A055496 A105120
Adjacent sequences: A083326 A083327 A083328 * A083330 A083331 A083332
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Apr 27 2003
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EXTENSIONS
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The generating function corrected by Martin Griffiths (griffm(AT)essex.ac.uk), Dec 01 2009
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STATUS
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approved
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