A005061 a(n) = 4^n - 3^n.

0, 1, 7, 37, 175, 781, 3367, 14197, 58975, 242461, 989527, 4017157, 16245775, 65514541, 263652487, 1059392917, 4251920575, 17050729021, 68332056247, 273715645477, 1096024843375, 4387586157901, 17560804984807, 70274600998837, 281192547174175, 1125052618233181

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of 2 X n binary arrays with a path of adjacent 1's from top row to bottom row, see A359576. - 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

If 2^k | n then a(2^k) | a(n). - Bernard Schott, Oct 08 2020

a(n) is the number of ordered n-tuples with elements from {0,1,2,3} in which any of these elements, say 0, appears at least once. For example, a(2)=7 since 01,10,02,20,03,30,00 are the ordered 2-tuples that contain 0. - Enrique Navarrete, Apr 05 2021

a(n) is the number of n-digit numbers whose smallest decimal digit is 6. - Stefano Spezia, Nov 15 2023

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• David Applegate, Marc 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]
• Dominique Désérable, Versatile Topology for Two-Dimensional Cellular Automata, Advances in Cellular Automata, Emergence, Complexity and Computation (ECC Vol 52) Springer, Cham (2025), Ch. 6, pp. 151-186.
• John Elias, Illustration of initial terms: Unfolded Sierpinski triangle or 3-branches tree in square configuration
• Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
• Vladeta Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (Russian, translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
• Eric Weisstein's World of Mathematics, Power Fractional Parts
• Index entries for linear recurrences with constant coefficients, signature (7,-12).
• Index entries for sequences related to dismal (or lunar) arithmetic

Crossrefs

Cf. A001047, A002250, A005060, A005062, A143495, A255463 (first differences), A359576.

Array column A047969(n-1, 3), or triangle's subdiagonal A047969(n+2, n-1), for n >= 1.

Programs

GAP

List([0..10^2], n->4*n - 3^n); # Muniru A Asiru, Feb 06 2018

Magma

[4^n - 3^n: n in [0..25]]; // Vincenzo Librandi, Jun 03 2011

Maple

seq(4^n - 3^n, n=0..10^2); # Muniru A Asiru, Feb 06 2018

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 *)
Table[Numerator[1-(3/4)^n],{n,0,20}] (* see link Wolfram Mathworld - Fred Daniel Kline, Feb 05 2018 *)

PARI

a(n)=1<<(n+n)-3^n \\ Charles R Greathouse IV, Jun 16 2011

Python

def a(n): return 4**n - 3**n
print([a(n) for n in range(23)]) # Michael S. Branicky, Sep 01 2021

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

From Mohammad K. Azarian, Jan 14 2009: (Start)

G.f.: 1/(1-4*x)-1/(1-3*x).

E.g.f.: exp(4*x)-exp(3*x). (End)

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

From Joe Slater, Jan 15 2017: (Start)

a(n) = 3*a(n-1) + 4^(n-1) for n>=0.

a(n+1) = Sum_{k=0..n} 4^(n-k) * 3^k. (End)

a(n) = -a(-n) * 12^n for all n in Z. - Michael Somos, Jan 22 2017