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%I #15 Feb 05 2015 12:04:53
%S 0,0,0,1,3,7,13,22,34,48,62,81,99,121,143,170,196,226,256,291,325,363,
%T 401,444,486,532,578,629,679,733,787,846,904,966,1028,1095,1161,1231,
%U 1301,1376,1450,1528,1606,1689,1771,1857,1943,2034,2124,2218,2312,2411,2509,2611
%N Total number of either concave decagons or concave hexadecagons 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 only an expansion obeying "two sides separated by one side" or one by obeying "two vertices separated by one vertex" for the "side-to-side" or "vertex-to-vertex" versions, respectively.
%C Two star-shaped hexadecagons (16-gons) and a 4-star appear when n = 8 for the "side-to-side" version, and in the "vertex-to-vertex" version there appears an irregular star-shaped icositetragons (24-gons). Rare type of polygons also appear for n > 8. See illustrations.
%H Kival Ngaokrajang, <a href="/A253896/a253896_2.pdf">Illustration of initial terms</a>, <a href="/A253896/a253896_3.pdf">Rare type polygons</a>
%F Empirical g.f.: -x^4*(2*x^10 -4*x^9 +2*x^8 -2*x^7 +2*x^5 +2*x^4 +2*x^3 +2*x^2 +x +1) / ((x -1)^3*(x +1)*(x^2 +1)). - _Colin Barker_, Jan 17 2015
%o (PARI)
%o {
%o a=0;d1=0;p=1;print1("0, 0, 0, ",p,", ");\\8s2a1
%o for(n=2,100,
%o if(n<5,d1=2,
%o if(n<7,d1=3,
%o if(n<8,d1=2,
%o if(Mod(n,4)==0,d1=0,
%o if(Mod(n,4)==1,d1=5,
%o if(Mod(n,4)==2,d1=-1,d1=4
%o )
%o )
%o )
%o )
%o )
%o );
%o a=a+d1;p=p+a;
%o print1(p,", ")
%o )
%o }
%Y Cf. A253895, 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 1,5
%A _Kival Ngaokrajang_, Jan 17 2015
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