

A337641


Onequarter of the number of regions in the central square of an equalarmed cross with arms of length n (as in A331456).


2



1, 14, 70, 231, 576, 1207, 2255, 3883, 6267, 9588, 14088, 20021, 27667, 37306, 49240, 63859, 81517, 102603, 127545, 156769, 190739, 229932, 274898, 326181, 384332, 449878, 523472, 605766, 697380, 799053, 911449, 1035371, 1171471, 1320566, 1483374, 1660873, 1853819, 2063133, 2289607, 2534326, 2798159
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OFFSET

0,2


COMMENTS

Without loss of generality, we may assume the central square has vertices (0,0), (1,0), (0,1), (1,1).
Suppose n >= 1. Then all nodes in the graph, whether or not in the central square, have rational coordinates with denominator at most 4*n^2 + 2*n + 1, which for n = 1, 2, 3, ... is 7, 21, 43, 73, 111, ... (cf. A054569).
This maximum is always attained, for example by the node at the intersection of the lines 2*n*x + y = n, joining (0,n) to (1, n) and x + (2*n+1)*y = n, joining (n,0) to (n+1,1).
In the central square, the maximum number of sides in any region is (for n = 0, 1, 2, 3, ...) 3, 4, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 6, 7, 6, 7, 7, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 6, 6, 6, 7, 7, 6, 6, 6, ... We conjecture that 7 is the maximum.  Lars Blomberg, Sep 19 2020.
See A331456 for further illustrations.


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CROSSREFS



KEYWORD

nonn,more


AUTHOR



STATUS

approved



