This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A174420 #19 Jun 25 2024 11:18:08
%S A174420 -2,-2,-1,0,8,4,26,98,68,245,284,941,908,2921,866,3038,9773,95842,
%T A174420 26864,82811,776048,235984,715436,2157533,14878043,27882168,16575521,
%U A174420 116892244,82326503,515542801,1009949246,882651721,500902958,1503356036,4511038850,13534572662,40605902153,53101505973,21870478820,65618808017
%N 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.
%C A174420 Related to Waring's problem.
%C A174420 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.
%C A174420 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
%D A174420 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 337.
%D A174420 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.
%H A174420 M. Waldschmidt, <a href="http://arXiv.org/abs/math.NT/0312440">Open Diophantine problems</a>
%t A174420 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 *)
%o A174420 (Python)
%o A174420 def A174420(n): return ((m:=3**n)&-(k:=1<<n))+k-m-(m>>n)-2 # _Chai Wah Wu_, Jun 25 2024
%Y A174420 Cf. A002804.
%K A174420 sign
%O A174420 0,1
%A A174420 _N. J. A. Sloane_, Nov 27 2010