A337641 - OEIS

%I #52 Jan 02 2024 12:46:05

%S 1,14,70,231,576,1207,2255,3883,6267,9588,14088,20021,27667,37306,

%T 49240,63859,81517,102603,127545,156769,190739,229932,274898,326181,

%U 384332,449878,523472,605766,697380,799053,911449,1035371,1171471,1320566,1483374,1660873,1853819,2063133,2289607,2534326,2798159

%N One-quarter of the number of regions in the central square of an equal-armed cross with arms of length n (as in A331456).

%C Without loss of generality, we may assume the central square has vertices (0,0), (1,0), (0,1), (1,1).

%C 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).

%C 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).

%C 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.

%C See A331456 for further illustrations.

%H Lars Blomberg, <a href="/A337641/b337641.txt">Table of n, a(n) for n = 0..74</a>

%H Scott R. Shannon, <a href="/A331452/a331452_6.png">Colored illustration for a(0)</a>: there are 4 regions, so a(0) = 1.

%H Scott R. Shannon, <a href="/A331456/a331456.png">Colored illustration for a(1)</a>: there are 56 regions, so a(1) = 14.

%H Scott R. Shannon, <a href="/A331456/a331456_1.png">Colored illustration for a(2)</a>: there are 280 regions, so a(2) = 70.

%H Scott R. Shannon, <a href="/A331456/a331456_2.png">Colored illustration for a(3)</a>: there are 924 regions, so a(3) = 231.

%H Scott R. Shannon, <a href="/A337641/a337641.png">Black and white illustration for a(1)</a> (Shows vertices and regions for each square)

%H Scott R. Shannon, <a href="/A337641/a337641_1.png">Black and white illustration for a(2)</a> (Shows vertices and regions for each square)

%H Scott R. Shannon, <a href="/A337641/a337641_2.png">Black and white illustration for a(3)</a> (Shows vertices and regions for each square)

%H Scott R. Shannon, <a href="/A337641/a337641_3.png">Black and white illustration for a(4)</a> (Shows vertices and regions for each square)

%H Scott R. Shannon, <a href="/A337641/a337641_4.png">Black and white illustration for a(5)</a> (Shows vertices and regions for each square)

%H Scott R. Shannon, <a href="/A337641/a337641_5.png">Black and white illustration for a(6)</a> (Shows vertices and regions for each square)

%Y Cf. A054569, A331456, A337640.

%K nonn

%O 0,2

%A _Lars Blomberg_, _Scott R. Shannon_, and _N. J. A. Sloane_, Sep 17 2020