A016177 a(n) = 8^n - 7^n.

0, 1, 15, 169, 1695, 15961, 144495, 1273609, 11012415, 93864121, 791266575, 6612607849, 54878189535, 452866803481, 3719823438255, 30436810578889, 248242046141055, 2019169299698041, 16385984911571535, 132716292890482729, 1073129238309234975, 8664826172771491801

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of ways to assign truth values to n ternary conjunctions connected by disjunctions such that the proposition is true. For example, a(2) = 15, since for the proposition '(a & b & c) v (d & e & f)' there are 15 assignments that make the proposition true. - Ori Milstein, Dec 22 2022

Equivalently, the number of length-n words over the alphabet {0,1,...,7} with at least one letter = 7. - Joerg Arndt, Jan 01 2023

a(n) is also the number of n-digit numbers whose smallest decimal digit is 2. - Stefano Spezia, Nov 15 2023

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (15,-56).

Crossrefs

Cf. A000420, A001018, A016140.

Programs

Magma

[8^n -7^n: n in [0..40]]; // G. C. Greubel, Nov 29 2024

Mathematica

Table[8^n - 7^n, {n, 0, 20}] (* Harvey P. Dale, Jan 31 2011 *)

PARI

a(n)=8^n-7^n \\ Charles R Greathouse IV, Sep 28 2015

Python

def A016177(n): return pow(8,n) - pow(7,n)
print([A016177(n) for n in range(41)]) # G. C. Greubel, Nov 29 2024

Formula

G.f.: x/((1-7x)*(1-8x)).

a(n) = numerator(f(n-1)) where f(n) = Integral_{x=0..1/4} (1-x/2)^n dx. And denominator(f(n)) = 4*(n+1)*8^n. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004 [corrected by Michel Marcus, Dec 23 2022]

a(n) = 15*a(n-1) - 56*a(n-2), n > 1. - Philippe Deléham, Jan 01 2009

E.g.f.: e^(8*x) - e^(7*x). - Mohammad K. Azarian, Jan 14 2009

a(n) = 8*a(n-1) + 7^(n-1), a(0)=0. - Vincenzo Librandi, Feb 09 2011