A005060 a(n) = 5^n - 4^n.

0, 1, 9, 61, 369, 2101, 11529, 61741, 325089, 1690981, 8717049, 44633821, 227363409, 1153594261, 5835080169, 29443836301, 148292923329, 745759583941, 3745977788889, 18798608421181, 94267920012849

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also, the number of numbers with at most n digits whose largest digit equals 4. - M. F. Hasler, May 03 2015

a(n) is divisible by 7 iff n is divisible by 6; for example: a(6) = 11529 = 7 * 1647 (see 'Les cahier du bac' or subtract A070365 and A153727 and locate zeros). - Bernard Schott, Oct 02 2020

a(n) is the number of n-digit numbers whose smallest decimal digit is 5. - Stefano Spezia, Nov 15 2023

References

• Les Cahiers du Bac, Terminales C & E, Tome 1, 1985, Exercice 109, p. 18; Bac Rouen, Série C, 1978.

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..200
• X. Acloque, Polynexus Numbers and other mathematical wonders [broken link]
• Samuele Giraudo, Pluriassociative algebras I: The pluriassociative operad, arXiv:1603.01040 [math.CO], 2016.
• Index entries for linear recurrences with constant coefficients, signature (9,-20).

Crossrefs

Cf. A070365, A153727.

Programs

GAP

List([0..20],n->5^n - 4^n); # Muniru A Asiru, Mar 04 2018

Maple

a:=n->sum(4^(n-j)*binomial(n,j),j=1..n): seq(a(n), n=0..18); # Zerinvary Lajos, Jan 04 2007

Mathematica

a[n_]:=5^n-4^n; a[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
LinearRecurrence[{9,-20},{0,1},30] (* Harvey P. Dale, Oct 01 2016 *)

PARI

a(n)=5^n-4^n \\ M. F. Hasler, May 03 2015

Sage

[lucas_number1(n, 9, 20) for n in range(21)]  # Zerinvary Lajos, Apr 23 2009

Formula

a(n) = 5*a(n-1) + 4^(n-1). - Xavier Acloque, Oct 20 2003

From Mohammad K. Azarian, Jan 14 2009: (Start)

G.f.: 1/(1-5*x) - 1/(1-4*x).

E.g.f.: e^(5*x) - e^(4*x). (End)

a(n) = 9*a(n-1) - 20*a(n-2), a(0)=0, a(1)=1. - Vincenzo Librandi, Jan 28 2011