A048329 Numbers that are repdigits in base 4.

0, 1, 2, 3, 5, 10, 15, 21, 42, 63, 85, 170, 255, 341, 682, 1023, 1365, 2730, 4095, 5461, 10922, 16383, 21845, 43690, 65535, 87381, 174762, 262143, 349525, 699050, 1048575, 1398101, 2796202, 4194303, 5592405, 11184810, 16777215, 22369621, 44739242, 67108863

Offset: 0

Author

Patrick De Geest, Feb 15 1999

Keywords

• nonn
• base
• easy

Examples

			10_10 = 22_4, 15_10 = 33_4, 5461_10 = 1111111_4.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Eric Weisstein's World of Mathematics, Repdigit.
• Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-4).

Crossrefs

Cf. A010785, A033017, A028987, A028988, A248721.

Base 4 repdigits 1,2,3 (trisections): A002450, A020988, A024036.

Programs

Magma

[0] cat  [k:k in [1..10^7]| #Set(Intseq(k,4)) eq 1]; // Marius A. Burtea, Oct 11 2019

Maple

a:= n-> (1+irem(n+2, 3))*(4^iquo(n+2,3)-1)/3:
seq(a(n), n = 0..45);

Mathematica

Union[Flatten[Table[FromDigits[PadRight[{}, n, d], 4], {n, 0, 40}, {d, 3}]]](* Vincenzo Librandi, Feb 06 2014 *)
LinearRecurrence[{0,0,5,0,0,-4},{0,1,2,3,5,10},40] (* Harvey P. Dale, Jul 11 2023 *)

Formula

G.f.: x*(1+2*x+3*x^2) / ( (x-1)*(4*x^3-1)*(1+x+x^2) ) with a(n) = 5*a(n-3) - 4*a(n-6). - R. J. Mathar, Mar 15 2015

Sum_{n>=1} 1/a(n) = (11/2) * A248721 = 2.31603727318383077512... - Amiram Eldar, Jan 21 2022