A016185 a(n) = 9^n - 8^n.

0, 1, 17, 217, 2465, 26281, 269297, 2685817, 26269505, 253202761, 2413042577, 22791125017, 213710059745, 1992110014441, 18478745943857, 170706760005817, 1571545212141185, 14425381885981321, 132080236787517137, 1206736529597136217, 11004743954450081825

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is also the number of n-digit numbers whose smallest decimal digit is 1. - Stefano Spezia, Nov 15 2023

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• John Elias, Illustration of initial terms: Sierpinski Carpet
• Wikipedia, Sierpinski carpet
• Index entries for linear recurrences with constant coefficients, signature (17,-72).

Crossrefs

Cf. A001018, A001019, A016140.

Programs

Magma

[9^n -8^n: n in [0..40]]; // G. C. Greubel, Nov 29 2024

Mathematica

LinearRecurrence[{17,-72}, {0,1}, 31] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)

Python

def A016185(n): return pow(9,n) - pow(8,n)
print([A016185(n) for n in range(41)]) # G. C. Greubel, Nov 29 2024

Formula

G.f.: x/((1-8*x)*(1-9*x)).

E.g.f.: e^(9*x) - e^(8*x). - Mohammad K. Azarian, Jan 14 2009

a(n) = 9*a(n-1) + 8^(n-1), a(0)=0. - Vincenzo Librandi, Feb 09 2011

a(n) = 17*a(n-1) - 72*a(n-2), a(0)=0, a(1)=1. - Vincenzo Librandi, Feb 09 2011

a(n) = A001019(n) - A001018(n). - Alois P. Heinz, Aug 19 2021