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A037314
Numbers whose base-3 and base-9 expansions have the same digit sum.
15
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
Also, every term k corresponds to a unique pair i,j with k = a(i) + 3*a(j) (similarly to the Moser-de Bruijn sequence). - Luis Rato, May 02 2024
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).
Sequence in context: A135782 A281899 A037457 * A226841 A218560 A373261
KEYWORD
nonn,base
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