A032858 - OEIS

%I #25 Oct 12 2018 21:18:17

%S 0,1,2,3,6,7,10,11,19,20,23,30,33,34,57,60,61,69,70,91,92,100,101,104,

%T 172,173,181,182,185,208,209,212,273,276,277,300,303,304,312,313,516,

%U 519,520,543,546,547,555,556,624,627,628,636,637

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

%C Every other base-3 digit must be strictly less than its neighbors. - _M. F. Hasler_, Oct 05 2018

%C 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

%H M. F. Hasler, <a href="/A032858/b032858.txt">Table of n, a(n) for n = 1..5000</a>

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

%e 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, ...

%t 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 *)

%o (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

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

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

%Y Cf. A306105 for these terms written in base 3.

%K nonn,base

%O 1,3

%A _Clark Kimberling_

%E Definition edited, cross-references and a(1) = 0 inserted by _M. F. Hasler_, Oct 05 2018