

A005061


a(n) = 4^n  3^n.


66



0, 1, 7, 37, 175, 781, 3367, 14197, 58975, 242461, 989527, 4017157, 16245775, 65514541, 263652487, 1059392917, 4251920575, 17050729021, 68332056247, 273715645477, 1096024843375, 4387586157901, 17560804984807
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OFFSET

0,3


COMMENTS

Number of 2 X n binary arrays with a path of adjacent 1's from top row to bottom row.  R. H. Hardin, 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_{2n1}, x_{2n}) both x_{2i1} 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_{2n1}*x_{2n} = 1 in base2 lunar 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.  Geoffrey Critzer, May 09 2009
This sequence is also the second column of the Sheffer triangle A143495 (3restricted Stirling2 numbers). (See the e.g.f. given below.)  _Wolfdieter Lang, Oct 08 2011
Also, the number of numbers with at most n digits whose largest digit equals 3. See A255463 for the first differences (i.e., ...with exactly n digits...).  M. F. Hasler, May 03 2015
For n > 1, each a(n) is a term in A116640.  Joe Slater, Jan 15 2017


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic"  the old name was too depressing]
Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
Index entries for linear recurrences with constant coefficients, signature (7,12).
Index entries for sequences related to dismal (or lunar) arithmetic


FORMULA

a(n) = 4*a(n1) + 3^(n1) for n>=1.  Xavier Acloque, Oct 20 2003
Binomial transform of A001047.  Ross La Haye, Sep 17 2005
G.f.: 1/(14*x)1/(13*x). E.g.f.: exp(4*x)exp(3*x).  Mohammad K. Azarian, Jan 14 2009
a(n) = 2^n * Sum_{i=0...n} binomial(n,i)*(2^i1)/2^i.  Geoffrey Critzer, May 09 2009
a(n) = 7*a(n1)  12*a(n2) for n>=2.  Bruno Berselli, Jan 25 2011
a(n) = 3*a(n1) + 4^(n1) for n>=0.  Joe Slater, Jan 15 2017
a(n+1) = Sum_{k=0..n} 4^(nk) * 3^k.  Joe Slater, Jan 15 2017
a(n) = a(n) * 12^n for all n in Z.  Michael Somos, Jan 22 2017


EXAMPLE

G.f. = x + 7*x^2 + 37*x^3 + 175*x^4 + 781*x^5 + 3367*x^6 + 14197*x^7 + ...


MATHEMATICA

Table[4^n  3^n, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Dec 21 2008 *)
LinearRecurrence[{7, 12}, {0, 1}, 30] (* Harvey P. Dale, May 04 2012 *)


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


CROSSREFS

Cf. A002250, A005060, A005062, A143495, A255463.
Sequence in context: A169726 A172063 A208737 * A099454 A177414 A125317
Adjacent sequences: A005058 A005059 A005060 * A005062 A005063 A005064


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



