A037314 Numbers whose base-3 and base-9 expansions have the same digit sum.

0, 1, 2, 9, 10, 11, 18, 19, 20, 81, 82, 83, 90, 91, 92, 99, 100, 101, 162, 163, 164, 171, 172, 173, 180, 181, 182, 729, 730, 731, 738, 739, 740, 747, 748, 749, 810, 811, 812, 819, 820, 821, 828, 829, 830, 891, 892, 893, 900, 901, 902, 909, 910, 911

Offset

0,3

Comments

a(n) = Sum_{i=0..m} d(i)*9^i, where Sum_{i=0..m} d(i)*3^i is the base-3 representation of n.

Numbers that can be written using only digits 0, 1 and 2 in base 9. Also, write n in base 3, read as base 9: (3) [n] (9) in base change notation. a(3n+k) = 9a(n)+k for k in {0,1,2}. - Franklin T. Adams-Watters, Jul 24 2006

Links

Table of n, a(n) for n=0..53.

Formula

G.f. f(x) = Sum_{j>=0} 9^j*x^(3^j)*(1+x^(3^j)-2*x^(2*3^j))/((1-x)*(1-x^(3^(j+1)))) satisfies f(x) = 9*(x^2+x+1)*f(x^3) + x*(1+2*x)/(1-x^3). - Robert Israel, Apr 13 2015

Mathematica

Table[FromDigits[RealDigits[n, 3], 9], {n, 1, 100}]
(* Clark Kimberling, Aug 14 2012 *)
Select[Range[0, 1000], Total[IntegerDigits[#, 3]]==Total[IntegerDigits[#, 9]]&] (* Harvey P. Dale, Feb 17 2020 *)

Prog

(PARI) a(n) = {my(d = digits(n, 3)); subst(Pol(d), x, 9); } \\ Michel Marcus, Apr 09 2015
(Julia)
function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 3)
        r += b * q
        b *= 9
    end
r end
[a(n) for n in 0:53] |> println # Peter Luschny, Jan 03 2021

Crossrefs

Cf. A007089, A208665, A338086 (ternary digit duplication).

Cf. A053735, A053830.

Sequence in context: A135782 A281899 A037457 * A226841 A218560 A031443

Adjacent sequences: A037311 A037312 A037313 * A037315 A037316 A037317

Keyword

nonn,base

Author

Clark Kimberling

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

Offset changed to 0 by Clark Kimberling, Aug 14 2012

Status

approved