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A002379 a(n) = floor(3^n / 2^n).
(Formerly M0666 N0245)
87

%I M0666 N0245

%S 1,1,2,3,5,7,11,17,25,38,57,86,129,194,291,437,656,985,1477,2216,3325,

%T 4987,7481,11222,16834,25251,37876,56815,85222,127834,191751,287626,

%U 431439,647159,970739,1456109,2184164,3276246,4914369,7371554,11057332

%N a(n) = floor(3^n / 2^n).

%C It is an important unsolved problem related to Waring's problem to show that a(n) = floor((3^n-1)/(2^n-1)) holds for all n > 1. This has been checked for 10000 terms and is true for all sufficiently large n, by a theorem of Mahler. [Lichiardopol]

%C a(n) = floor((3^n-1)/(2^n-1)) holds true at least for 2 <= n <= 305000. - _Hieronymus Fischer_, Dec 31 2008

%C a(n) is also the curve length (rounded down) of the Sierpiński arrowhead curve after n iterations, let a(0) = 1. - _Kival Ngaokrajang_, May 21 2014

%C a(n) is composite infinitely often (Forman and Shapiro). More exactly, a(n) is divisible by at least one of 2, 5, 7 or 11 infinitely often (Dubickas and Novikas). - _Tomohiro Yamada_, Apr 15 2017

%D R. K. Guy, Unsolved Problems in Number Theory, E19.

%D D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 82.

%D S. S. Pillai, On Waring's problem, J. Indian Math. Soc., 2 (1936), 16-44.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002379/b002379.txt">Table of n, a(n) for n = 0..1000</a>

%H Arturas Dubickas, Aivaras Novikas, <a href="https://doi.org/10.1007/s00209-005-0827-4">Integer parts of powers of rational numbers</a>, Math. Z. 251 (2005), 635--648, available from <a href="http://www.mif.vu.lt/~dubickas/files/dvifai/suaivaru.pdf">the first author's page</a>.

%H W. Forman and H. N. Shapiro, <a href="https://doi.org/10.1002/cpa.3160200305">An arithmetic property of certain rational powers</a>, Comm. Pure. Appl. Math. 20 (1967), 561-573.

%H R. K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20.

%H R. K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]

%H N. Lichiardopol, <a href="http://dx.doi.org/10.1016/j.disc.2007.07.021">Problem 925 (BCC20.19)</a>, A number-theoretic problem, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.

%H K. Mahler, <a href="http://dx.doi.org/10.1112/S0025579300001170">On the fractional parts of the powers of a rational number, II</a>, Mathematika 4 (1957), 122-124.

%H Kival Ngaokrajang, <a href="/A002379/a002379_1.pdf">Illustration of Sierpinski arrowhead curve for n = 0..5</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerFloors.html">Power Floors</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sierpi%C5%84ski_arrowhead_curve">Sierpiński arrowhead curve</a>

%F a(n) = b(n) - (-2/3)^n where b(n) is defined by the recursion b(0):=2, b(1):=5/6, b(n+1):=(5/6)*b(n) + b(n-1). - _Hieronymus Fischer_, Dec 31 2008

%F a(n) = (1/2)*(b(n) + sqrt(b(n)^2 - (-4)^n)) (with b(n) as defined above). - _Hieronymus Fischer_, Dec 31 2008

%F 3^n = a(n)*2^n + A002380(n). - _R. J. Mathar_, Oct 26 2012

%F a(n) = -(1/2) + (3/2)^n + arctan(cot((3/2)^n Pi)) / Pi. - _Fred Daniel Kline_, Apr 14 2018

%F a(n+1) = round( -(1/2) + (3^n-1)/(2^n-1) ). - _Fred Daniel Kline_, Apr 14 2018

%p A002379:=n->floor(3^n/2^n); seq(A002379(k), k=0..100); # _Wesley Ivan Hurt_, Oct 29 2013

%t Table[Floor[(3/2)^n], {n, 0, 40}] (* _Robert G. Wilson v_, May 11 2004 *)

%t x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi; Array[x, 40] (* _Fred Daniel Kline_, Dec 21 2017 *)

%t x[n_]:=Round[-(1/2) + (3^n - 1)/(2^n - 1)]; Array[x, 39, 2] (* offset n+1, _Fred Daniel Kline_, Apr 13 2018 *)

%o (PARI) a(n)=3^n>>n \\ _Charles R Greathouse IV_, Jun 10 2011

%o (MAGMA) [Floor(3^n / 2^n): n in [0..40]]; // _Vincenzo Librandi_, Sep 08 2011

%o (Maxima) makelist(floor(3^n/2^n), n, 0, 50); /* _Martin Ettl_, Oct 17 2012 */

%o (Haskell)

%o a002379 n = 3^n `div` 2^n -- _Reinhard Zumkeller_, Jul 11 2014

%Y Cf. A094969-A094500, A000217, A081464, A153662, A153665, A153666.

%Y Cf. A060692, A002380, A000079, A000244.

%Y Cf. A046037, A070758, A070759, A067904 (Composites and Primes).

%Y Cf. A064628 (an analog for 4/3).

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

%E More terms from _Robert G. Wilson v_, May 11 2004

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Last modified November 18 17:57 EST 2019. Contains 329288 sequences. (Running on oeis4.)