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A005061 a(n) = 4^n - 3^n. 65
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; text; internal format)
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_{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 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 (3-restricted 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, 127-138 (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(n-1) + 3^(n-1) for n>=1. - Xavier Acloque, Oct 20 2003

Binomial transform of A001047. - Ross La Haye, Sep 17 2005

G.f.: 1/(1-4*x)-1/(1-3*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^i-1)/2^i. - Geoffrey Critzer, May 09 2009

a(n) = 7*a(n-1) - 12*a(n-2) for n>=2. - Bruno Berselli, Jan 25 2011

a(n) = 3*a(n-1) + 4^(n-1) for n>=0. - Joe Slater, Jan 15 2017

a(n+1) = Sum_{k=0..n} 4^(n-k) * 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

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Last modified August 19 01:36 EDT 2017. Contains 290788 sequences.