A253896 -id:A253896 - cp's OEIS frontend

A253895 Total number of octagons in two variants of an octagon expansion after n iterations: either "side-to-side" or "vertex-to-vertex", respectively.

1, 3, 7, 14, 25, 41, 63, 90, 120, 154, 192, 233, 278, 328, 382, 439, 500, 566, 636, 709, 786, 868, 954, 1043, 1136, 1234, 1336, 1441, 1550, 1664, 1782, 1903, 2028, 2158, 2292, 2429, 2570, 2716, 2866, 3019, 3176, 3338, 3504, 3673, 3846, 4024, 4206, 4391, 4580, 4774, 4972

Offset: 1

Kival Ngaokrajang, Jan 17 2015

nonn

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 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.
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.

Kival Ngaokrajang, Illustration of initial terms, Rare type polygons

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).

{
a=1;d1=0;p=a;print1(a,", ");\\8s2a, total oct.
for(n=2,100,
   if(n<=7,d1=n-1,
     if(n<9,d1=5,
       if(n<10,d1=3,
         if(n<11,d1=4,
           if(Mod(n,4)==0,d1=3,
             if(Mod(n,4)==1,d1=4,
               if(Mod(n,4)==2,d1=5,d1=4
               )
             )
           )
         )
       )
     )
   );
   a=a+d1;p=p+a;
   print1(p,", ")
)
}

Conjectures from Colin Barker, Jan 17 2015: (Start)
a(n) = (-4-i*(-i)^n+i*i^n-18*n+8*n^2)/4 for n>8, where i=sqrt(-1).
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)).
(End)

A254835 Total number of nonagons in a variant of a nonagon expansion ("side-to-side", two consecutive sides) after n iterations.

Original entry on oeis.org

2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 133, 136, 144, 153, 161, 170, 180, 187, 197, 206, 216, 225, 233, 242, 248, 259, 269, 278, 286, 295, 305, 314, 322, 331, 341, 350, 358, 367, 377, 386, 394, 403, 413, 422, 430, 439, 449, 458, 466, 475, 485, 494, 502

Offset: 1

Kival Ngaokrajang, Feb 08 2015

nonn

Two irregular star-shaped 18-gons appear for n = 17.

There are also rare types of polygons appearing for n >= 16. See illustrations.

Kival Ngaokrajang, Illustration of initial terms, Rare type polygons

Cf. 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), A253895 and A253896 (Octagon expansion).

{a=259;print1("2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 133, 136, 144, 153, 161, 170, 180, 187, 197, 206, 216, 225, 233, 242, 248, ",a,", "); for(n=32,100,if(Mod(n,4)==0,d=10,if(Mod(n,4)==1,d=9,if(Mod(n,4)==2, d=8, d=9)));a=a+d;print1(a,", "))}

Conjectures from Colin Barker, Feb 08 2015: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>21.
G.f.: -x*(2*x^33 -4*x^32 +4*x^31 -6*x^30 +4*x^29 +2*x^26 -4*x^25 +4*x^24 -4*x^23 +2*x^22 -2*x^20 -4*x^19 +8*x^18 -2*x^17 +8*x^16 +2*x^15 -2*x^14 -2*x^13 -2*x^12 -2*x^11 -2*x^10 -2*x^9 -2*x^8 -2*x^7 -2*x^6 -2*x^5 -2*x^4 -x^3 -3*x^2 -2) / ((x -1)^2*(x^2 +1)).
(End)

cp's OEIS Frontend

A253895 Total number of octagons in two variants of an octagon expansion after n iterations: either "side-to-side" or "vertex-to-vertex", respectively.

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