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A007283
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a(n) = 3*2^n.
(Formerly M2561)
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163
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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
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OFFSET
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0,1
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COMMENTS
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Same as Pisot sequences E(3,6), L(3,6), P(3,6), T(3,6). See A008776 for definitions of Pisot sequences.
Numbers n such that P[phi[n]]=phi[P[n]], where P[x] is the largest prime-factor of x; A006530[A000010(n)]=A000010[A006530(n)]=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 first differences are the 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 n-dimensional hypercubes (Latin polyhedra) of order 3. - Kenji Ohkuma (k-ookuma(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
3 times powers of 2. - Omar E. Pol, Dec 16 2008
From Jacobsthal numbers and successive differences. A001045=0,1,1,3,5,11,21,43,; A078008=1,0,2,2,6,10,22,42,; A084247 signed=-1,2,0,4,4,12,20,44,; A154879=3,-2,4,0,8,8,24,40,. For every sequence k(n), a(n)= k(n)+k(n+3).Then A001045(n)+A001045(n+3), A078008(n)+A078008(n+3), .. . - Paul Curtz, Feb 05 2009
a(n) written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, n times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009
a(n) = A173786(n+1,n) = A173787(n+2,n). - Reinhard Zumkeller, Feb 28 2010
Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010
Numbers containing the number 3 in their Collatz trajectories. - Reinhard Zumkeller, Feb 20 2012
For n > 0: A216022(a(n)) = 6 and A216059(a(n)) = 7. - Reinhard Zumkeller, Sep 01 2012
a(n-1) 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) = 3x-4. This equation also has solutions 84, 3348, 1450092, ... which are not of the form 3*2^n. - Farideh Firoozbakht, Nov 30 2013
a(n) = upper bound for the "X-ray 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^(n-2) 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 real-valued 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 k-point subsets S in E. Moreover, the best possible k is 3*2^(n-1).
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
A020651(a(n)) = 2. - Yosu Yurramendi, Jun 01 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
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REFERENCES
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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 JP2004-272104A, 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, Information-Technology Promotion Agency, JAPAN.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
K. Bezdek and T. Zamfirescu, A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
Yuri Brudnyi, and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
T. Doslic, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
Tanya Khovanova, Recursive Sequences
E. Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
Index entries for linear recurrences with constant coefficients, signature (2).
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FORMULA
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G.f.: 3/(1-2*x).
a(n) = 2*a(n-1), n>0; a(0)=3.
a(n) = Sum_{k=0..n} (-1)^(k reduced (mod 3))*C(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
a(n) = 2^n + 2^(n+1). - Jaroslav Krizek, Aug 17 2009
a(n) = (A000225(n)+1)*3. - Martin Ettl, Nov 11 2012
E.g.f: 3*exp(2*x). - Ilya Gutkovskiy, May 15 2016
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MAPLE
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A007283:=n->3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013
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MATHEMATICA
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Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)
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PROG
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(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 */
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CROSSREFS
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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.
Sequence in context: A170636 A170684 A003945 * A049942 A200463 A099844
Adjacent sequences: A007280 A007281 A007282 * A007284 A007285 A007286
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KEYWORD
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easy,nonn
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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