A253547 Total number of star-shaped dodecagons appearing in a variant of hexagon expansion after n iterations.

0, 0, 0, 1, 3, 9, 16, 23, 33, 43, 56, 69, 85, 101, 120, 139, 161, 183, 208, 233, 261, 289, 320, 351, 385, 419, 456, 493, 533, 573, 616, 659, 705, 751, 800, 849, 901, 953, 1008, 1063, 1121, 1179, 1240, 1301, 1365, 1429, 1496, 1563, 1633, 1703, 1776, 1849, 1925, 2001, 2080

Offset: 1

Author

Kival Ngaokrajang, Jan 03 2015

Keywords

• nonn
• nice
• easy

Comments

Total number of hexagons after n iterations is A179178. See illustration.

Links

• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A179178.

Programs

Mathematica

LinearRecurrence[{2,0,-2,1},{0,0,0,1,3,9,16,23,33},60] (* Harvey P. Dale, Oct 30 2015 *)

PARI

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

Formula

Conjectures from Colin Barker, Jan 03 2015: (Start)

a(n) = (27 - 3*(-1)^n - 28*n + 6*n^2)/8 for n>5.

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

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