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

1,5

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

Table of n, a(n) for n=1..54.

Kival Ngaokrajang, Illustration of initial terms, Rare type polygons

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

Prog

(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, ", ")
)
}

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).

Sequence in context: A136219 A078582 A051336 * A002623 A173196 A081662

Adjacent sequences: A253893 A253894 A253895 * A253897 A253898 A253899

Keyword

nonn

Author

Kival Ngaokrajang, Jan 17 2015

Status

approved