A048333 Numbers that are repdigits in base 8.

0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54, 63, 73, 146, 219, 292, 365, 438, 511, 585, 1170, 1755, 2340, 2925, 3510, 4095, 4681, 9362, 14043, 18724, 23405, 28086, 32767, 37449, 74898, 112347, 149796, 187245, 224694, 262143, 299593, 599186, 898779

Offset: 0

Author

Patrick De Geest, Feb 15 1999

Keywords

• nonn
• base
• easy

Comments

For the general case, the sequence of numbers that are repdigits in base b > 1 satisfies the recurrence a(n) = (b+1)*a(n-b+1) - b*a(n-2*(b-1)) for n >= 2(b-1) with g.f.: (sum_{1 <= i < b} i*x^i)/(1 - (b+1)*x^(b-1) + bx^(2(b-1))). - Chai Wah Wu, May 30 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..700
• Eric Weisstein's World of Mathematics, Repdigit.
• Index entries for 2-automatic sequences.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,9,0,0,0,0,0,0,-8).

Crossrefs

Cf. A010785, A033021, A028987, A028988, A248725.

Programs

Mathematica

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 8], {n, 0, 40}, {d, 7}]]] (* Vincenzo Librandi, Feb 06 2014 *)
LinearRecurrence[{0,0,0,0,0,0,9,0,0,0,0,0,0,-8},{0,1,2,3,4,5,6,7,9,18,27,36,45,54},50] (* Harvey P. Dale, Dec 09 2018 *)

PARI

is(n)=#Set(digits(n,8))==1 \\ Charles R Greathouse IV, Feb 15 2017

Formula

From Chai Wah Wu, May 30 2016: (Start)

a(n) = 9*a(n-7) - 8*a(n-14) for n > 13.

G.f.: x*(7*x^6 + 6*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1)/(8*x^14 - 9*x^7 + 1). (End)

Sum_{n>=1} 1/a(n) = (363/20) * A248725 = 2.92153624531838250201... - Amiram Eldar, Jan 21 2022

Extensions

Changed offset from 1 to 0 by Vincenzo Librandi, Feb 06 2014