A253688 The total number of pentagons in a variant of pentagon expansion (vertex-to-vertex, two consecutive vertices and one isolated vertex) after n iterations.

1, 4, 10, 21, 39, 64, 94, 129, 171, 218, 272, 331, 397, 468, 546, 629, 719, 814, 916, 1023, 1137, 1256, 1382, 1513, 1651, 1794, 1944, 2099, 2261, 2428, 2602, 2781, 2967, 3158, 3356, 3559, 3769, 3984, 4206, 4433, 4667, 4906, 5152, 5403, 5661, 5924, 6194, 6469, 6751, 7038

Offset: 1

Author

Kival Ngaokrajang, Jan 09 2015

Keywords

• nonn

Comments

Two star shaped icosagons appearing at n >= 6. See illustration.

Links

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A253687 (side-to-side).

Programs

PARI

{
a=1;d1=0;p=a;print1(a,", ");\\5v3b
for(n=2,100,
   if(n<3,d1=2,
     if(n<4,d1=3,
       if(n<5,d1=5,
         if(n<6,d1=7,
           if(n<7,d1=7,
             if(n<8,d1=5,
               if(Mod(n,2)==0,d1=5,d1=7
               )
             )
           )
         )
       )
     )
   );
   a=a+d1;p=p+a;
   print1(p,", ")
)
}

Formula

Conjectures from Colin Barker, Jan 09 2015: (Start)

a(n) = (53-(-1)^n-38*n+12*n^2)/4 for n>5.

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>9.

G.f.: -x*(2*x^8-2*x^7-2*x^6+2*x^5+4*x^4+3*x^3+2*x^2+2*x+1) / ((x-1)^3*(x+1)).

(End)