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%I #19 Dec 24 2015 18:26:57
%S 1,3,7,14,25,41,63,90,120,154,192,233,278,328,382,439,500,566,636,709,
%T 786,868,954,1043,1136,1234,1336,1441,1550,1664,1782,1903,2028,2158,
%U 2292,2429,2570,2716,2866,3019,3176,3338,3504,3673,3846,4024,4206,4391,4580,4774,4972
%N Total number of octagons in two variants of an octagon expansion after n iterations: either "side-to-side" or "vertex-to-vertex", respectively.
%C Inspired by A061777 and A179178 which are "vertex-to-vertex" and "side-to-side" versions of equilateral triangle expansion respectively.
%C In these octagon expansions there is allowed an expansion obeying "two sides separated by one side" or one obeying "two vertices separated by one vertex" for "side-to-side" or "vertex-to-vertex" versions respectively.
%C Two star shaped hexadecagons (16-gons) and a 4-star appear for n = 8 in the "side-to-side" version, and in the "vertex-to-vertex" version there appear two irregular star shaped icositetragons (24-gons). There are also rare type of polygons appearing for n > 8. See illustrations.
%H Kival Ngaokrajang, <a href="/A253895/a253895_2.pdf">Illustration of initial terms</a>, <a href="/A253895/a253895_3.pdf">Rare type polygons</a>
%F Conjectures from _Colin Barker_, Jan 17 2015: (Start)
%F a(n) = (-4-i*(-i)^n+i*i^n-18*n+8*n^2)/4 for n>8, where i=sqrt(-1).
%F G.f.: -x*(x^12-2*x^10-x^8+2*x^6+2*x^5+2*x^4+x^3+2*x^2+1) / ((x-1)^3*(x^2+1)).
%F (End)
%o (PARI)
%o {
%o a=1;d1=0;p=a;print1(a,", ");\\8s2a, total oct.
%o for(n=2,100,
%o if(n<=7,d1=n-1,
%o if(n<9,d1=5,
%o if(n<10,d1=3,
%o if(n<11,d1=4,
%o if(Mod(n,4)==0,d1=3,
%o if(Mod(n,4)==1,d1=4,
%o if(Mod(n,4)==2,d1=5,d1=4
%o )
%o )
%o )
%o )
%o )
%o )
%o );
%o a=a+d1;p=p+a;
%o print1(p,", ")
%o )
%o }
%Y Cf. A253896, A061777 (Triangle expansion, vertex-to-vertex, 3 vertices), A179178 (Triangle expansion, side-to-side, 2 sides), A253687 (Pentagon expansion, side-to-side, 2 consecutive sides and 1 isolated side), A253688 (Pentagon expansion, vertex-to-vertex, 2 consecutive vertices and 1 isolated vertex), A253547 (Hexagon expansion, vertex-to-vertex, 2 vertices separated by 1 vertex).
%K nonn
%O 1,2
%A _Kival Ngaokrajang_, Jan 17 2015
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