A005062 a(n) = 6^n - 5^n.

0, 1, 11, 91, 671, 4651, 31031, 201811, 1288991, 8124571, 50700551, 313968931, 1932641711, 11839990891, 72260648471, 439667406451, 2668522016831, 16163719991611, 97745259402791, 590286253682371

Offset

0,3

Comments

These are the numerators of a(n) = (Integral_{x=0..1/3} (1-x/2)^n dx). E.g., a(3)=671/2592. The denominators are b(n) = 3*(n+1)*6^n. E.g., b(3)=2592. the subscripts in both cases are 0. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004

Number of numbers with at most n digits whose largest digit is 5. For the first 5 terms, the first differences (i.e., ...with exactly n digits...) are given in A125373. - M. F. Hasler, May 03 2015

a(n) is the number of n-digit numbers whose smallest decimal digit is 4. - Stefano Spezia, Nov 15 2023

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Index entries for linear recurrences with constant coefficients, signature (11,-30).

Formula

G.f.: x/((1-5*x)(1-6*x)).

a(n) = 11*a(n-1) - 30*a(n-2), n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009

E.g.f.: exp(6*x) - exp(5*x). - Mohammad K. Azarian, Jan 14 2009

a(n) = -(30)^n * a(-n) for all n in Z. - Michael Somos, Jul 14 2018

Example

G.f. = x + 11*x^2 + 91*x^3 + 671*x^4 + 4651*x^5 + 31031*x^6 + 201811*x^7 + ... - Michael Somos, Jul 14 2018

Maple

restart:a:=n->sum(5^(n-j)*binomial(n, j), j=1..n): seq(a(n), n=0..19); # Zerinvary Lajos, Apr 18 2009

Mathematica

f[n_]:=6^n-5^n; f[Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
LinearRecurrence[{11, -30}, {0, 1}, 20] (* Harvey P. Dale, May 28 2015 *)

Prog

(Sage) [lucas_number1(n, 11, 30) for n in range(0, 20)] # Zerinvary Lajos, Apr 27 2009
(Magma) [6^n - 5^n: n in [0..25]]; // Vincenzo Librandi, Jun 03 2011
(PARI) a(n)=6^n-5^n \\ M. F. Hasler, May 03 2015
(PARI) for(d=0, 9, print1(sum(n=1, 10^d-1, vecmax(digits(n))==5)", ")) \\ Only to illustrate the comment about "largest digit equals 5".

Crossrefs

Cf. A005060 (5^n - 4^n), A125373.

Sequence in context: A201085 A055083 A016160 * A125374 A245599 A126532

Adjacent sequences: A005059 A005060 A005061 * A005063 A005064 A005065

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved