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%I #28 Sep 19 2015 18:57:16
%S 1,3,9,12,24,12,24,18,36,33,57,45,81,36,78,42,90,57,111,69,135,60,132,
%T 66,144,81,165,93,189,84,186,90,198,105,219,117,243,108,240,114,252,
%U 129,273,141,297,132,294,138,306,153
%N Start with a single equilateral triangle for n=0; for the odd n-th generation add a triangle at each expandable side of the triangles of the (n-1)-th generation (this is the "vertex to side" version); for the even n-th generation use the "side to side" version; a(n) is the number of triangles added in the n-th generation.
%C See a comment on V-V and V=S at A249246.
%C The overlap rules for the expansion are: (i) overlap within generation is allowed. (ii) overlap of different generations is prohibited.
%C There are a total of 16 combinations as shown in the table below:
%C +-------------------------------------------------------+
%C | Even n-th version V-V S-V V-S S-S |
%C +-------------------------------------------------------+
%C | Odd n-th version |
%C | V-V A008486 A248969 A261951 A261952 |
%C | S-V A261950 A008486 A008486 A261956 |
%C | V-S A249246 A008486 A008486 a(n) |
%C | S-S A261953 A261954 A261955 A008486 |
%C +-------------------------------------------------------+
%C Note: V-V = vertex to vertex, S-V = side to vertex,
%C V-S = vertex to side, S-S = side to side.
%H Kival Ngaokrajang, <a href="/A261957/a261957.pdf">Illustration of initial terms</a>
%F Conjectures from _Colin Barker_, Sep 10 2015: (Start)
%F a(n) = a(n-2)+a(n-8)-a(n-10) for n>14.
%F G.f.: (3*x^14+9*x^13-9*x^12-3*x^11-13*x^10-12*x^9-11*x^8-6*x^7-15*x^4-9*x^3-8*x^2-3*x-1) / ((x-1)^2*(x+1)^2*(x^2+1)*(x^4+1)).
%F (End)
%o (PARI) {e=24; o=12; print1("1, 3, 9, 12, 24, ", o, ", ", e, ", "); for(n=7, 100, if (Mod(n,2)==0, if (Mod(n,8)==0, e=e+12); if (Mod(n,8)==2, e=e+21); if (Mod(n,8)==4, e=e+24); if (Mod(n,8)==6, e=e-3); print1(e, ", "), if (Mod(n,8)==7, o=o+6); if (Mod(n,8)==1, o=o+15); if (Mod(n,8)==3, o=o+12); if (Mod(n,8)==5, o=o-9); print1(o, ", ")))}
%Y Cf. A008486, A248969, A249246.
%K nonn
%O 0,2
%A _Kival Ngaokrajang_, Sep 06 2015
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