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A356984 Number of regions in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.

%I A356984 #70 Dec 06 2022 15:51:55
%S A356984 1,4,13,28,49,70,109,148,181,244,301,334,433,508,565,676,769,811,973,
%T A356984 1069,1165,1324,1453,1534,1729,1876,1957,2182,2353,2446,2701,2884,
%U A356984 3013,3268,3454,3538,3889,4108,4261,4519,4801,4960,5293,5536,5668,6076,6349,6502,6913,7204,7405,7798,8113
%N A356984 Number of regions in an equilateral triangle when n internal equilateral triangles are drawn between the 3n points that divide each side into n+1 equal parts.
%C A356984 See A357007 for further images.
%H A356984 Scott R. Shannon, <a href="/A356984/b356984.txt">Table of n, a(n) for n = 0..250</a>
%H A356984 Scott R. Shannon, <a href="/A356984/a356984.jpg">Image for n = 1</a>.
%H A356984 Scott R. Shannon, <a href="/A356984/a356984_1.jpg">Image for n = 2</a>.
%H A356984 Scott R. Shannon, <a href="/A356984/a356984_2.jpg">Image for n = 3</a>.
%H A356984 Scott R. Shannon, <a href="/A356984/a356984_3.jpg">Image for n = 5</a>. This is the first term that forms intersections with non-simple vertices.
%H A356984 Scott R. Shannon, <a href="/A356984/a356984_4.jpg">Image for n = 10</a>.
%H A356984 Scott R. Shannon, <a href="/A356984/a356984_5.jpg">Image for n = 50</a>.
%H A356984 Scott R. Shannon, <a href="/A356984/a356984_6.jpg">Image for n = 100</a>.
%H A356984 Scott R. Shannon, <a href="/A356984/a356984_7.jpg">Image for n = 200</a>.
%H A356984 Talmon Silver, <a href="/A356984/a356984.rtf">Classification of the intersection points and the number of regions</a>
%F A356984 a(n) = A357008(n) - A357007(n) + 1 by Euler's formula.
%F A356984 Conjecture: a(n) = 3*n^2 + 1 for equilateral triangles that only contain simple vertices when cut by n internal equilateral triangles. This is never the case if (n + 1) mod 3 = 0 for n > 3.
%F A356984 a(n) = 1 + 3*n + T2(n) + 2*T3(n) + 3*T4(n); a(n) = 1 + 3*n^2 - T3(n) - 3*T4(n), where T2 is the number of internal vertices meeting exactly two segments (these vertices are labeled in the A357007 links as "4 ngons"), T3 is the number of internal vertices meeting exactly three segments ("6 ngons"), and T4 is the number of internal vertices meeting exactly four segments ("8 ngons"). - _Talmon Silver_, Sep 23 2022
%Y A356984 Cf. A357007 (vertices), A357008 (edges), A092867, A092098, A332953, A343755.
%K A356984 nonn
%O A356984 0,2
%A A356984 _Scott R. Shannon_, Sep 08 2022