login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A174420 Let A(n) = floor((3/2)^n), B(n)=3^n-2^n*A(n); then a(n)=2^n-A(n)-B(n)-2. 1
-2, -2, -1, 0, 8, 4, 26, 98, 68, 245, 284, 941, 908, 2921, 866, 3038, 9773, 95842, 26864, 82811, 776048, 235984, 715436, 2157533, 14878043, 27882168, 16575521, 116892244, 82326503, 515542801, 1009949246, 882651721, 500902958, 1503356036, 4511038850, 13534572662, 40605902153, 53101505973, 21870478820, 65618808017 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Related to Waring's problem.
a(n) is conjectured to be >= 0 for all n >= 3. If this were proved it would settle Waring's problem (see A002804). It is known that a(n) >= 0 for 3 <= n <= 471600000.
If we rewrite the formula as (2^n-1)*ceiling((3/2)^n) - 3^n - 1, we see more clearly a comparison between 3^n and the product of an undervaluation of 2^n and an overvaluation of (3/2)^n. If the undervaluation is proportionately smaller than the ceiling overvaluation, a(n) is nonnegative. 2^n grows faster than (3/2)^n, so for a negative value to occur the target difference between (3/2)^n and ceiling((3/2)^n) gets smaller as n gets larger, and the sum of these target differences (for n > 0) is finite. - Peter Munn, Dec 08 2022
REFERENCES
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 337.
R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
LINKS
MATHEMATICA
a[n_] := Floor[(3/2)^n]; b[n_] := 3^n - 2^n*a[n]; Table[2^n - a[n] - b[n] - 2, {n, 0, 39}] (* Jean-François Alcover, Apr 05 2011 *)
CROSSREFS
Cf. A002804.
Sequence in context: A327365 A093729 A113080 * A360604 A266318 A011265
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Nov 27 2010
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)