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0, 1, 7, 37, 175, 781, 3367, 14197, 58975, 242461, 989527, 4017157, 16245775, 65514541, 263652487, 1059392917, 4251920575, 17050729021, 68332056247, 273715645477, 1096024843375, 4387586157901, 17560804984807
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of 2 X n binary arrays with a path of adjacent 1's from top row to bottom row. - R. H. Hardin (rhhardin(AT)att.net), Mar 21 2002
Number of binary vectors (x_1, x_2, ..., x_{2n}) such that in at least one of the disjoint pairs (x_1, x_2), (x_3, x_4), ..., (x_{2n-1}, x_{2n}) both x_{2i-1} and x_{2i} are both 1. Equivalently, number of solutions (x_1, ..., x_n) to the equation x_1*x_2 + x_3*x_4 + x_5*x_6 + ... +x_{2n-1}*x_{2n} = 1 in base 2 dismal arithmetic. - N. J. A. Sloane, Apr 23 2011.
a(n)/4^n is the probability that two randomly selected (with replacement) subsets of [n] will have at least one element in common if the probability of selection is equal for all subsets. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 09 2009]
This sequence is also the second column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). See the e.g.f. given below.) [Wolfdieter Lang, Oct 08 2011]
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REFERENCES
| V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..300
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic
Index to sequences with linear recurrences with constant coefficients, signature (7,-12).
Index entries for sequences related to dismal arithmetic
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FORMULA
| a(n) = 4*a(n-1) + 3^(n-1) - Xavier Acloque Oct 20 2003
Binomial transform of A001047. - Ross La Haye (rlahaye(AT)new.rr.com), Sep 17 2005
G.f.: 1/(1-4*x)-1/(1-3*x). E.g.f.: exp(4*x)-exp(3*x). [From Mohammad K. Azarian (azarian(AT)evansville.edu), Jan 14 2009]
a(n) = 2^n * sum(i=0...n, binomial(n,i)*(2^i-1)/2^i ) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 09 2009]
a(n) = +7*a(n-1)-12*a(n-2) for n>=2. - Bruno Berselli, Jan 25 2011
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MATHEMATICA
| Table[4^n - 3^n, {n, 0, 20}] (* From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 21 2008 *)
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PROG
| (MAGMA) [4^n - 3^n: n in [0..25]]; // Vincenzo Librandi, Jun 03 2011
(PARI) a(n)=1<<(n+n)-3^n \\ Charles R Greathouse IV, Jun 16 2011
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CROSSREFS
| Cf. A002250.
Sequence in context: A169789 A169726 A172063 * A099454 A177414 A125317
Adjacent sequences: A005058 A005059 A005060 * A005062 A005063 A005064
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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