A241275 a(n) = 6*a(n-1) + 5*(2^(n-1)-1) for n > 0, a(0) = 0.

0, 0, 5, 45, 305, 1905, 11585, 69825, 419585, 2518785, 15115265, 90696705, 544190465, 3265163265, 19591020545, 117546205185, 705277394945, 4231664697345, 25389988839425, 152339934347265, 914039608705025, 5484237657473025, 32905425955323905, 197432555752914945, 1184595334559432705, 7107572007440482305

Offset: 0

Author

Kival Ngaokrajang, Apr 18 2014

Keywords

• nonn
• easy

Comments

a(n) is the total number of holes in a pentaflake after n iterations. The number of irregular polygon-shaped holes is 5*A016200(n-3), n >= 3. The number of rhombic-shaped holes is 5*A016129(n-2), n >= 2 and equal to the first difference of a(n).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Crossrefs

Cf. A016129, A016200, A240523.

Programs

Maple

A241275:=n->9*6^(n-2) - 5*2^(n-2) + 1; seq(A241275(n), n=0..40); # Wesley Ivan Hurt, Apr 19 2014

Mathematica

CoefficientList[Series[-5 x^2/((x - 1) (2 x - 1) (6 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 19 2014 *)
LinearRecurrence[{9,-20,12},{0,0,5},30] (* Harvey P. Dale, Oct 10 2017 *)

PARI

a(n)=if(n<=0, 0, a(n-1)*6+5*(2^(n-1)-1))
       for(n=0, 100, print1(a(n),", "))

Formula

a(n) = 5*(A016129(n-2) + A016200(n-3)) for n >= 3, a(0)=a(1)=0, a(2)=5.

a(n) = 5*A016200(n-2) for n >= 2.

a(n) = 9*6^(n-2) - 5*2^(n-2) + 1.

a(n) = (4-5*2^n+6^n)/4. G.f.: -5*x^2 / ((x-1)*(2*x-1)*(6*x-1)). - Colin Barker, Apr 18 2014

Extensions

Typo in DATA fixed by Colin Barker, Apr 18 2014