0,3

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 (Hieronymus.Fischer(AT)gmx.de), Dec 31 2008

R. K. Guy, The second strong law of small numbers. Math. Mag. 63 (1990), no. 1, 3-20.

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

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.

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).

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

Eric Weisstein's World of Mathematics, Power Floors

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 (Hieronymus.Fischer(AT)gmx.de), Dec 31 2008

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

Table[ Floor[(3/2)^n], {n, 0, 40}] (from Robert G. Wilson v)

Cf. A094969 - A094500.

Cf. A000217, A153661, A153662, A153665, A153666.

Sequence in context: A068523 A055500 A018058 this_sequence A072465 A052284 A133670

Adjacent sequences: A002376 A002377 A002378 this_sequence A002380 A002381 A002382

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 11 2004