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A253896 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.

%I A253896 #15 Feb 05 2015 12:04:53
%S A253896 0,0,0,1,3,7,13,22,34,48,62,81,99,121,143,170,196,226,256,291,325,363,
%T A253896 401,444,486,532,578,629,679,733,787,846,904,966,1028,1095,1161,1231,
%U A253896 1301,1376,1450,1528,1606,1689,1771,1857,1943,2034,2124,2218,2312,2411,2509,2611
%N A253896 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 A253896 Inspired by A061777 and A179178 which are "vertex-to-vertex" and "side-to-side" versions of equilateral triangle expansion, respectively.
%C A253896 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 A253896 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 A253896 Kival Ngaokrajang, <a href="/A253896/a253896_2.pdf">Illustration of initial terms</a>, <a href="/A253896/a253896_3.pdf">Rare type polygons</a>
%F A253896 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 A253896 (PARI)
%o A253896 {
%o A253896 a=0;d1=0;p=1;print1("0, 0, 0, ",p,", ");\\8s2a1
%o A253896 for(n=2,100,
%o A253896    if(n<5,d1=2,
%o A253896      if(n<7,d1=3,
%o A253896        if(n<8,d1=2,
%o A253896          if(Mod(n,4)==0,d1=0,
%o A253896            if(Mod(n,4)==1,d1=5,
%o A253896              if(Mod(n,4)==2,d1=-1,d1=4
%o A253896              )
%o A253896            )
%o A253896          )
%o A253896        )
%o A253896      )
%o A253896    );
%o A253896    a=a+d1;p=p+a;
%o A253896    print1(p,", ")
%o A253896 )
%o A253896 }
%Y A253896 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 A253896 nonn
%O A253896 1,5
%A A253896 _Kival Ngaokrajang_, Jan 17 2015