

A253896


Total number of either concave decagons or concave hexadecagons in two variants of an octagon expansion after n iterations: either "sidetoside" or "vertextovertex", respectively.


3



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
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OFFSET

1,5


COMMENTS

Inspired by A061777 and A179178 which are "vertextovertex" and "sidetoside" 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 "sidetoside" or "vertextovertex" versions, respectively.
Two starshaped hexadecagons (16gons) and a 4star appear when n = 8 for the "sidetoside" version, and in the "vertextovertex" version there appears an irregular starshaped icositetragons (24gons). Rare type of polygons also appear for n > 8. See illustrations.


LINKS



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, vertextovertex, 3 vertices), A179178 (Triangle expansion, sidetoside, 2 sides), A253687 (Pentagon expansion, sidetoside, 2 consecutive sides and 1 isolated side), A253688 (Pentagon expansion, vertextovertex, 2 consecutive vertices and 1 isolated vertex), A253547 (Hexagon expansion, vertextovertex, 2 vertices separated by 1 vertex).


KEYWORD

nonn


AUTHOR



STATUS

approved



