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

1, 1, 2, 3, 5, 7, 11, 17, 25, 38, 57, 86, 129, 194, 291, 437, 656, 985, 1477, 2216, 3325, 4987, 7481, 11222, 16834, 25251, 37876, 56815, 85222, 127834, 191751, 287626, 431439, 647159, 970739, 1456109, 2184164, 3276246, 4914369, 7371554, 11057332

Offset

0,3

Comments

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]

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

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

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

References

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

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

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

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

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

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Arturas Dubickas and Aivaras Novikas, Integer parts of powers of rational numbers, Math. Z. 251 (2005), 635--648, available from the first author's page.

W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers, Comm. Pure. Appl. Math. 20 (1967), 561-573.

R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.

R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]

N. Lichiardopol, Problem 925 (BCC20.19), A number-theoretic problem, in Research Problems from the 20th British Combinatorial Conference, Discrete Math., 308 (2008), 621-630.

K. Mahler, On the fractional parts of the powers of a rational number, II, Mathematika 4 (1957), 122-124.

Kival Ngaokrajang, Illustration of Sierpinski arrowhead curve for n = 0..5

Eric Weisstein's World of Mathematics, Power Floors

Wikipedia, Sierpiński arrowhead curve

Formula

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

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

3^n = a(n)*2^n + A002380(n). - R. J. Mathar, Oct 26 2012

a(n) = -(1/2) + (3/2)^n + arctan(cot((3/2)^n Pi)) / Pi. - Fred Daniel Kline, Apr 14 2018

a(n+1) = round( -(1/2) + (3^n-1)/(2^n-1) ). - Fred Daniel Kline, Apr 14 2018

Maple

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

Mathematica

Table[Floor[(3/2)^n], {n, 0, 40}] (* Robert G. Wilson v, May 11 2004 *)
x[n_] := -(1/2) + (3/2)^n + ArcTan[Cot[(3/2)^n Pi]]/Pi; Array[x, 40] (* Fred Daniel Kline, Dec 21 2017 *)
x[n_]:=Round[-(1/2) + (3^n - 1)/(2^n - 1)]; Array[x, 39, 2] (* offset n+1, Fred Daniel Kline, Apr 13 2018 *)

Prog

(PARI) a(n)=3^n>>n \\ Charles R Greathouse IV, Jun 10 2011
(Magma) [Floor(3^n / 2^n): n in [0..40]]; // Vincenzo Librandi, Sep 08 2011
(Maxima) makelist(floor(3^n/2^n), n, 0, 50); /* Martin Ettl, Oct 17 2012 */
(Haskell)
a002379 n = 3^n `div` 2^n  -- Reinhard Zumkeller, Jul 11 2014
(Python)
def A002379(n): return 3**n>>n # Chai Wah Wu, Sep 21 2022

Crossrefs

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

Cf. A060692, A002380, A000079, A000244.

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

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

Sequence in context: A345020 A353505 A018058 * A072465 A204631 A323361

Adjacent sequences: A002376 A002377 A002378 * A002380 A002381 A002382

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Robert G. Wilson v, May 11 2004

Status

approved