cp's OEIS Frontend

A182298 Smallest complementary perimeter, as defined in the comments, among all sets of nonnegative integers whose volume (sum) is n.

%I A182298 #36 Nov 16 2023 15:56:41
%S A182298 0,2,4,3,6,5,4,7,7,6,5,10,8,8,7,6,12,11,9,9,8,7,11,13,12,10,10,9,8,15,
%T A182298 12,14,13,11,11,10,9,17,16,13,15,14,12,12,11,10,17,18,17,14,16,15,13,
%U A182298 13,12,11,16,18,19,18,15,17,16,14,14,13,12,21,17,19,20
%N A182298 Smallest complementary perimeter, as defined in the comments, among all sets of nonnegative integers whose volume (sum) is n.
%C A182298 The volume and perimeter of a set S of nonnegative integers are introduced in the reference.  The volume is defined simply as the sum of the elements of S, and the perimeter is defined as the sum of the elements of S whose predecessor and successor are not both in S.  The complementary perimeter (introduced in the link) of S is the perimeter of the complement of S in the set of nonnegative integers.
%H A182298 Martin Ehrenstein, <a href="/A182298/b182298.txt">Table of n, a(n) for n = 0..250</a>
%H A182298 Patrick Devlin, <a href="http://arxiv.org/abs/1107.2954">Sets with High Volume and Low Perimeter</a>, arXiv:1107.2954 [math.CO], 2011.
%H A182298 Patrick Devlin, <a href="http://arxiv.org/abs/1202.1331">Integer Subsets with High Volume and Low Perimeter</a>, arXiv:1202.1331 [math.CO], 2012.
%H A182298 Patrick Devlin, <a href="http://www.emis.de/journals/INTEGERS/papers/m32/m32.Abstract.html">Integer Subsets with High Volume and Low Perimeter</a>, INTEGERS, Vol. 12, #A32.
%H A182298 J. Miller, F. Morgan, E. Newkirk, L. Pedersen and D. Seferis, <a href="http://www.jstor.org/stable/10.4169/math.mag.84.1.037">Isoperimetric Sets of Integers</a>, Math. Mag. 84 (2011) 37-42.
%F A182298 Following the notation in the link, for n >= 0, write n = (0+1+2+...+f(n)) - g(n), be the representation of n with f(n) and g(n) minimal such that 0 <= g(n) <= f(n).  Then f(n) = A002024(n) = round(sqrt(2n)), and g(n) = A025581(n) = f(n)*(f(n)+1)/2 - n.
%F A182298 Finally, let Q(n):=a(n), and let P(n):=A186053(n).  Then unless n is one of the 177 known counterexamples tabulated in the link, we have P(n) = f(n) + Q(g(n)), and Q(n) = 1 + f(n) + P(g(n)).
%e A182298 For n=8, the set S={0,1,3,4} has volume (total sum) 8 and complementary perimeter (the sum of 2 and 5) is 7.  No other set of volume 8 has a smaller complementary perimeter, so a(8)=7.
%e A182298 Similarly, for n=11, the set S={2,4,5} has volume 11=2+4+5 and complementary perimeter 10=1+3+6.  This is the smallest among all sets with volume 11, so a(11)=10.
%Y A182298 Cf. A186053.
%K A182298 nonn
%O A182298 0,2
%A A182298 _Patrick Devlin_, Apr 23 2012
%E A182298 More terms from _Martin Ehrenstein_, Nov 16 2023