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A182298
Smallest complementary perimeter, as defined in the comments, among all sets of nonnegative integers whose volume (sum) is n.
7
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
0,2
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.
LINKS
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.
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)).
EXAMPLE
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.
CROSSREFS
Cf. A186053.
Sequence in context: A175027 A269837 A308982 * A182732 A119477 A161920
KEYWORD
nonn
AUTHOR
Patrick Devlin, Apr 23 2012
EXTENSIONS
More terms from Martin Ehrenstein, Nov 16 2023
STATUS
approved