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

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, 12, 14, 13, 11, 11, 10, 9, 17, 16, 13, 15, 14, 12, 12, 11, 10, 17, 18, 17, 14, 16, 15, 13, 13, 12, 11, 16, 18, 19, 18, 15, 17, 16, 14, 14, 13, 12, 21, 17, 19, 20

Offset: 0

Author

Patrick Devlin, Apr 23 2012

Keywords

• nonn

Comments

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.

Examples

			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.
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.
		

Links

• Martin Ehrenstein, Table of n, a(n) for n = 0..250
• Patrick Devlin, Sets with High Volume and Low Perimeter, arXiv:1107.2954 [math.CO], 2011.
• Patrick Devlin, Integer Subsets with High Volume and Low Perimeter, arXiv:1202.1331 [math.CO], 2012.
• Patrick Devlin, Integer Subsets with High Volume and Low Perimeter, INTEGERS, Vol. 12, #A32.
• J. Miller, F. Morgan, E. Newkirk, L. Pedersen and D. Seferis, Isoperimetric Sets of Integers, Math. Mag. 84 (2011) 37-42.

Crossrefs

Cf. A186053.

Formula

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.

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)).

Extensions

More terms from Martin Ehrenstein, Nov 16 2023