

A007283


a(n) = 3*2^n.
(Formerly M2561)


208



3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

Same as Pisot sequences E(3, 6), L(3, 6), P(3, 6), T(3, 6). See A008776 for definitions of Pisot sequences.
Numbers k such that A006530(A000010(k)) = A000010(A006530(k)) = 2.  Labos Elemer, May 07 2002
Also least number m such that 2^n is the smallest proper divisor of m which is also a suffix of m in binary representation, see A080940.  Reinhard Zumkeller, Feb 25 2003
Length of the period of the sequence Fibonacci(k) (mod 2^(n+1)).  Benoit Cloitre, Mar 12 2003
The sequence of first differences is this sequence itself.  Alexandre Wajnberg and Eric Angelini, Sep 07 2005
Subsequence of A122132.  Reinhard Zumkeller, Aug 21 2006
Apart from the first term, a subsequence of A124509.  Reinhard Zumkeller, Nov 04 2006
Total number of Latin ndimensional hypercubes (Latin polyhedra) of order 3.  Kenji Ohkuma (kookuma(AT)ipa.go.jp), Jan 10 2007
Number of different ternary hypercubes of dimension n.  Edwin Soedarmadji (edwin(AT)systems.caltech.edu), Dec 10 2005
For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n + 1} > {1, 2, 3} such that for fixed, different x_1, x_2,...,x_n in {1, 2, ..., n + 1} and fixed y_1, y_2,...,y_n in {1, 2, 3} we have f(x_i) <> y_i, (i = 1,2,...,n).  Milan Janjic, May 10 2007
a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953).  Jaroslav Krizek, Aug 17 2009
Subsequence of A051916.  Reinhard Zumkeller, Mar 20 2010
Numbers containing the number 3 in their Collatz trajectories.  Reinhard Zumkeller, Feb 20 2012
a(n1) gives the number of ternary numbers with n digits with no two adjacent digits in common; e.g., for n=3 we have 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210 and 212.  Jon Perry, Oct 10 2012
If n > 1, then a(n) is a solution for the equation sigma(x) + phi(x) = 3x4. This equation also has solutions 84, 3348, 1450092, ... which are not of the form 3*2^n.  Farideh Firoozbakht, Nov 30 2013
a(n) is the upper bound for the "Xray number" of any convex body in E^(n + 2), conjectured by Bezdek and Zamfirescu, and proved in the plane E^2 (see the paper by Bezdek and Zamfirescu).  L. Edson Jeffery, Jan 11 2014
If T is a topology on a set V of size n and T is not the discrete topology, then T has at most 3 * 2^(n2) many open sets. See Brown and Stephen references.  Ross La Haye, Jan 19 2014
Comment from Charles Fefferman, courtesy of Doron Zeilberger, Dec 02 2014: (Start)
Fix a dimension n. For a realvalued function f defined on a finite set E in R^n, let Norm(f, E) denote the inf of the C^2 norms of all functions F on R^n that agree with f on E. Then there exist constants k and C depending only on the dimension n such that Norm(f, E) <= C*max{ Norm(f, S) }, where the max is taken over all kpoint subsets S in E. Moreover, the best possible k is 3 * 2^(n1).
The analogous result, with the same k, holds when the C^2 norm is replaced, e.g., by the C^1, alpha norm (0 < alpha <= 1). However, the optimal analogous k, e.g., for the C^3 norm is unknown.
For the above results, see Y. Brudnyi and P. Shvartsman (1994). (End)
Also, coordination sequence for (infinity, infinity, infinity) tiling of hyperbolic plane.  N. J. A. Sloane, Dec 29 2015
The average of consecutive powers of 2 beginning with 2^1.  Melvin Peralta and Miriam Ong Ante, May 14 2016
For n > 1, a(n) is the smallest Zumkeller number with n divisors that are also Zumkeller numbers (A083207).  Ivan N. Ianakiev, Dec 09 2016
Also, for n >= 2, the number of lengthn strings over the alphabet {0,1,2,3} having only the single letters as nonempty palindromic subwords. (Corollary 21 in Fleischer and Shallit)  Jeffrey Shallit, Dec 02 2019
Also, a(n) is the minimum linklength of any covering trail, circuit, path, and cycle for the set of the 2^(n+2) vertices of an (n+2)dimensional hypercube.  Marco Ripà, Aug 22 2022


REFERENCES

Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004272104A, Japan Patent Office (written in Japanese, a(2)=12, a(3)=24, a(4)=48, a(5)=96, a(6)=192, a(7)=384 (a(7)=284 was corrected)).
Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, InformationTechnology Promotion Agency, JAPAN.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. Bezdek and Tudor Zamfirescu, A Characterization of 3dimensional Convex Sets with an Infinite Xray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), NorthHolland, Amsterdam, 1991, pp. 3338.
Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129139.
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239257. See Prop. 3.1.
Tomislav Došlić, KeplerBouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
Tanya Khovanova, Recursive Sequences
Edwin Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 12321239
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739  741, 1968.
Index entries for linear recurrences with constant coefficients, signature (2).


FORMULA

G.f.: 3/(12*x).
a(n) = 2*a(n  1), n > 0; a(0) = 3.
a(n) = Sum_{k = 0..n} (1)^(k reduced (mod 3))*binomial(n, k).  Benoit Cloitre, Aug 20 2002
a(n) = A118416(n + 1, 2) for n > 1.  Reinhard Zumkeller, Apr 27 2006
a(n) = A000079(n) + A000079(n + 1).  Zerinvary Lajos, May 12 2007
a(n) = A000079(n)*3.  Omar E. Pol, Dec 16 2008
From Paul Curtz, Feb 05 2009: (Start)
a(n) = b(n) + b(n+3) for b = A001045, A078008, A154879.
a(n) = abs(b(n)  b(n+3)) with b(n) = (1)^n*A084247(n). (End)
a(n) = 2^n + 2^(n + 1).  Jaroslav Krizek, Aug 17 2009
a(n) = A173786(n + 1, n) = A173787(n + 2, n).  Reinhard Zumkeller, Feb 28 2010
A216022(a(n)) = 6 and A216059(a(n)) = 7, for n > 0.  Reinhard Zumkeller, Sep 01 2012
a(n) = (A000225(n) + 1)*3.  Martin Ettl, Nov 11 2012
E.g.f.: 3*exp(2*x).  Ilya Gutkovskiy, May 15 2016
A020651(a(n)) = 2.  Yosu Yurramendi, Jun 01 2016
a(n) = sqrt(A014551(n + 1)*A014551(n + 2) + A014551(n)^2).  Ezhilarasu Velayutham, Sep 01 2019
a(A048672(n)) = A225546(A133466(n)).  Michel Marcus and Peter Munn, Nov 29 2019
Sum_{n>=1} 1/a(n) = 2/3.  Amiram Eldar, Oct 28 2020


MAPLE

A007283:=n>3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013


MATHEMATICA

Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)


PROG

(PARI) a(n)=3*2^n
(PARI) a(n)=3<<n \\ Charles R Greathouse IV, Oct 10 2012
(Magma) [3*2^n: n in [0..30]]; // Vincenzo Librandi, May 18 2011
(Haskell)
a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
 Reinhard Zumkeller, Mar 18 2012, Feb 20 2012
(Maxima) A007283(n):=3*2^n$
makelist(A007283(n), n, 0, 30); /* Martin Ettl, Nov 11 2012 */
(Scala) (List.fill(40)(2: BigInt)).scanLeft(1: BigInt)(_ * _).map(3 * _) // Alonso del Arte, Nov 28 2019


CROSSREFS

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Subsequence of A029744.
Essentially same as A003945 and A042950.
Row sums of (5, 1)Pascal triangle A093562 and of (1, 5) Pascal triangle A096940.
Cf. A000079, A100540, A124508, A221718.
Cf. Latin squares: A000315, A002860, A003090, A040082, A003191; Latin cubes: A098843, A098846, A098679, A099321.
Cf. A133466, A225546.
Sequence in context: A170636 A170684 A003945 * A049942 A200463 A099844
Adjacent sequences: A007280 A007281 A007282 * A007284 A007285 A007286


KEYWORD

easy,nonn


AUTHOR

N. J. A. Sloane


STATUS

approved



