A016203 Expansion of g.f. 1/((1-x)*(1-2*x)*(1-8*x)).

1, 11, 95, 775, 6231, 49911, 399415, 3195575, 25565111, 204521911, 1636177335, 13089422775, 104715390391, 837723139511, 6701785148855, 53614281256375, 428914250182071, 3431314001718711, 27450512014273975, 219604096115240375, 1756832768924020151, 14054662151396355511

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

4*a(n) is the total number of holes in a certain box fractal (start with 8 boxes, 0 hole) after n iterations. See illustration in link. - Kival Ngaokrajang, Jan 27 2015

Links

• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (11,-26,16).

Crossrefs

Cf. A016131, A023001.

Programs

Maple

a:=n->sum((8^(n-j)-2^(n-j))/6,j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 15 2007

PARI

Vec(1/((1-x)*(1-2*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (4*8^(n+1) - 7*2^(n+1) + 3)/21. - Mitch Harris, Jun 27 2005; corrected by Yahia Kahloune, May 06 2013

a(0) = 1, a(n) = 8*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 09 2011

From Elmo R. Oliveira, Mar 26 2025: (Start)

E.g.f.: exp(x)*(32*exp(7*x) - 14*exp(x) + 3)/21.

a(n) = 11*a(n-1) - 26*a(n-2) + 16*a(n-3).

a(n) = A016131(n+1) - A023001(n+2). (End)

Extensions

More terms from Elmo R. Oliveira, Mar 26 2025