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A253895 Total number of octagons in two variants of an octagon expansion after n iterations: either "side-to-side" or "vertex-to-vertex", respectively. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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 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.

LINKS

Table of n, a(n) for n=1..51.

Kival Ngaokrajang, Illustration of initial terms, Rare type polygons

FORMULA

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)

PROG

(PARI)

{

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, ", ")

)

}

CROSSREFS

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).

Sequence in context: A316319 A179178 A171973 * A004006 A089240 A057524

Adjacent sequences:  A253892 A253893 A253894 * A253896 A253897 A253898

KEYWORD

nonn

AUTHOR

Kival Ngaokrajang, Jan 17 2015

STATUS

approved

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Last modified February 18 09:57 EST 2019. Contains 320250 sequences. (Running on oeis4.)