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A325946 Maximum number of intercardinal adjacencies among all n-celled polyplets.

%I A325946 #37 Jul 12 2021 22:37:02
%S A325946 0,1,3,6,8,11,14,17,20,23,26,30,33,36,39,43,46,49,53,56,60,63,66,70,
%T A325946 73,77,80,84,87,91,94,98,101,105,108,112,116,119,123,126,130,133,137,
%U A325946 141,144,148,151,155,159,162,166,170,173,177,180,184,188,191,195,199,202,206,210,213,217,221,224,228,232,235,239,243,246,250,254
%N A325946 Maximum number of intercardinal adjacencies among all n-celled polyplets.
%C A325946 A123663 provides the maximum number of cardinal adjacencies among n-celled polyominoes. The sequence under consideration here provides the maximum number of intercardinal (edge-to-edge and vertex-to-vertex) adjacencies among all n-celled polyplets.
%C A325946 Both A123663 and this sequence are used by landscape ecologists and geographic information system (GIS) professionals to determine quantitative measures over time of landscape erosion in high density coastal areas.
%C A325946 For initial terms n <= 20, M_O(n) is known to be optimal; for n > 20, the optimality of M_O(n) is probable. - _Nicholas P. Taliceo_, Jul 12 2021
%H A325946 K. McGarigal, <a href="http://www.umass.edu/landeco/research/fragstats/fragstats.html">FRAGSTATS Webpage—Contains the Aggregation Index</a>
%H A325946 N. P. Taliceo, <a href="https://nicholastaliceo.com/research/intercardinal-aggregation-index/icai-code-python3/">ICAI Python Code</a>
%H A325946 N. P. Taliceo, <a href="http://www.westfield.ma.edu/math/MathWeb/REU/PDFs/SHPAbstract.pdf">Intercardinal Adjacencies: A New Landscape Metric</a>, Westfield State University Honors Program (2016).
%H A325946 N. P. Taliceo and J. F. Fleron (2021), <a href="https://doi.org/10.1080/0025570X.2021.1843961">A Prime Example of the Strong Law of Small Numbers</a>, Mathematics Magazine, 94:1, 59-61.
%F A325946 By empirical observation a split-rule formula with 15 conditions generates the sequence M = 0, 1, 3, 6, 8, 11, ... correctly for small n - this includes comparison with configurations known to be optimal (n < 20) and with computer generated searches for optimal configurations (n < 500):
%F A325946 M_O(n) = 4n - 14p + 10 - e
%F A325946 where
%F A325946 n = number of tiles in the polyplet t
%F A325946 p = Max{p>=1 : n >= 7p^2-10p+4}
%F A325946 e = 0, if n = 7p^2 - 10p + 4
%F A325946 e = 1, if 7p^2 - 10p + 4 < n <= 7p^2 - 9p + 3
%F A325946 e = 2, if 7p^2 - 9p + 3 < n <= 7p^2 - 8p + 2
%F A325946 e = 3, if 7p^2 - 8p + 2 < n <= 7p^2 - 7p + 2
%F A325946 e = 4, if 7p^2 - 7p + 2 < n <= 7p^2 - 6p + 1
%F A325946 e = 5, if 7p^2 - 6p + 1 < n <= 7p^2 - 5p + 1
%F A325946 e = 6, if 7p^2 - 5p + 1 < n <= 7p^2 - 4p + 1
%F A325946 e = 7, if 7p^2 - 4p + 1 < n <= 7p^2 - 3p
%F A325946 e = 8, if 7p^2 - 3p < n <= 7p^2 - 2p
%F A325946 e = 9, if 7p^2 - 2p < n <= 7p^2 - p
%F A325946 e = 10, if 7p^2 - p < n <= 7p^2
%F A325946 e = 11, if 7p^2 < n <= 7p^2 + p
%F A325946 e = 12, if 7p^2 + p < n <= 7p^2 + 2p
%F A325946 e = 13, if 7p^2 + 2p < n <= 7p^2 + 3p
%F A325946 e = 14, if n > 7p^2 + 3p
%F A325946 This split-rule formula is derived geometrically using an approach described in the Example section.
%F A325946 Subsequently we have proved that M_O can be represented analytically by a single expression: M_O(n) = 4n-ceiling(sqrt(28n-12)).
%F A325946 We have proved the important estimate M_O(n) <= M(n) <= 2*(2n-2*ceiling(sqrt(n))) where 2n-2*ceiling(sqrt(n)) is A123663. This upper bound is not sharp for small n. The relative difference between M_O(n) and 2*(2n-2*ceiling(sqrt(n))) is less than 3% for at least 145 <= n <= 10^7 tiles and the relative difference goes to zero. For practical uses like GIS, our formula will have very small relative error if, in fact, it does not describe the sequence exactly.
%e A325946 For n = 12, the optimal configuration is a "regular octagon" of side length two (i.e., the symmetric, cross-shaped configuration with rows of length 2, 4, 4, and 2). This yields 30 intercardinal adjacencies.
%e A325946 In general, when n = 7p^2 - 10p + 4 the n tiles can be arranged into the shape of a regular octagon with side length p and 28p^2 - 54p + 26 intercardinal adjacencies. We conjecture these are optimal.
%e A325946 Moreover, we believe all of the intermediary cases are generated by a family of archetypes where one moves from a regular octagon to a "stretched octagon" to a "small corners octagon" and then to the next largest regular octagon. This geometric approach gives rise to the split rule formula described above.
%o A325946 (Python) # See N. P. Taliceo link.
%Y A325946 The Aggregation Index is cataloged as A123663.
%K A325946 nonn
%O A325946 1,3
%A A325946 _Nicholas P. Taliceo_ and _Julian F. Fleron_, Sep 09 2019