A032858 Numbers whose base-3 representation Sum_{i=0..m} d(i)*3^i has d(m) > d(m-1) < d(m-2) > ...

0, 1, 2, 3, 6, 7, 10, 11, 19, 20, 23, 30, 33, 34, 57, 60, 61, 69, 70, 91, 92, 100, 101, 104, 172, 173, 181, 182, 185, 208, 209, 212, 273, 276, 277, 300, 303, 304, 312, 313, 516, 519, 520, 543, 546, 547, 555, 556, 624, 627, 628, 636, 637

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base

Comments

Every other base-3 digit must be strictly less than its neighbors. - M. F. Hasler, Oct 05 2018

The terms can be generated in the following way: if A(n) are the terms with n digits in base 3, the terms with n+2 digits are obtained by prefixing them with '10' and with '20', and prefixing '21' to those starting with a digit '2'. It is easy to prove that #A(n) = A000045(n+2), since from the above we have #A(n+2) = 2*#A(n) + #A(n-1) = #A(n) + #A(n+1). (The #A(n-1) numbers starting with '2' are #A(n-2) numbers prefixed with '20' and #A(n-3) prefixed with '21'.) - M. F. Hasler, Oct 05 2018

Examples

			The base-3 representation of the initial terms is 0, 1, 2, 10, 20, 21, 101, 102, 201, 202, 212, 1010, 1020, 1021, 2010, 2020, 2021, 2120, 2121, 10101, 10102, ...
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A032859 .. A032865 for base-4 .. 10 variants.

Cf. A000975 (or A056830 in binary) for the base-2 analog.

Cf. A306105 for these terms written in base 3.

Programs

Mathematica

sdQ[n_]:=Module[{s=Sign[Differences[IntegerDigits[n, 3]]]}, s==PadRight[{}, Length[s], {-1, 1}]]; Select[Range[0, 700], sdQ] (* Vincenzo Librandi, Oct 06 2018 *)

PARI

is(n,b=3)=!for(i=2,#n=digits(n,b),(n[i-1]-n[i])*(-1)^i>0||return) \\ M. F. Hasler, Oct 05 2018

Formula

a(A000071(n+3)) = floor(3^(n+1)/8) = A033113(n). - M. F. Hasler, Oct 05 2018

Extensions

Definition edited, cross-references and a(1) = 0 inserted by M. F. Hasler, Oct 05 2018