A253687 The total number of pentagons in a variant of pentagon expansion (side-to-side, two consecutive sides and one isolated side) after n iterations.

1, 4, 10, 21, 39, 64, 94, 129, 167, 212, 262, 317, 375, 440, 510, 585, 663, 748, 838, 933, 1031, 1136, 1246, 1361, 1479, 1604, 1734, 1869, 2007, 2152, 2302, 2457, 2615, 2780, 2950, 3125, 3303, 3488, 3678, 3873, 4071, 4276, 4486, 4701, 4919, 5144, 5374, 5609, 5847, 6092

Offset: 1

Author

Kival Ngaokrajang, Jan 09 2015

Keywords

• nonn

Comments

Two decagons appearing at n >= 6. See illustration.

Links

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

Cf. A253688 (vertex-to-vertex).

Programs

PARI

{
a=1;d1=0;p=a;print1(a,", ");\\5s3b
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(Mod(n,4)==0,d1=5,
             if(Mod(n,4)==1,d1=3,
               if(Mod(n,4)==2,d1=7,d1=5
               )
             )
           )
         )
       )
     )
   );
   a=a+d1;p=p+a;
   print1(p,", ")
)
}

Formula

Conjecture: a(n) = 2*a(n-1)-a(n-2)+a(n-4)-2*a(n-5)+a(n-6) for n>7. - Colin Barker, Jan 09 2015

Empirical g.f.: x*(4*x^8-2*x^6-5*x^5-6*x^4-5*x^3-3*x^2-2*x-1) / ((x-1)^3*(x+1)*(x^2+1)). - Colin Barker, Jan 09 2015