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.

0, 0, 0, 1, 3, 7, 13, 22, 34, 48, 62, 81, 99, 121, 143, 170, 196, 226, 256, 291, 325, 363, 401, 444, 486, 532, 578, 629, 679, 733, 787, 846, 904, 966, 1028, 1095, 1161, 1231, 1301, 1376, 1450, 1528, 1606, 1689, 1771, 1857, 1943, 2034, 2124, 2218, 2312, 2411, 2509, 2611

Offset: 1

Author

Kival Ngaokrajang, Jan 17 2015

Keywords

• nonn

Comments

Inspired by A061777 and A179178 which are "vertex-to-vertex" and "side-to-side" versions of equilateral triangle expansion, respectively.

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.

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.

Links

• Kival Ngaokrajang, Illustration of initial terms, Rare type polygons

Crossrefs

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).

Programs

PARI

{
a=0;d1=0;p=1;print1("0, 0, 0, ",p,", ");\\8s2a1
for(n=2,100,
   if(n<5,d1=2,
     if(n<7,d1=3,
       if(n<8,d1=2,
         if(Mod(n,4)==0,d1=0,
           if(Mod(n,4)==1,d1=5,
             if(Mod(n,4)==2,d1=-1,d1=4
             )
           )
         )
       )
     )
   );
   a=a+d1;p=p+a;
   print1(p,", ")
)
}

Formula

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