cp's OEIS Frontend

A247648 Numbers whose binary expansion begins and ends with 1 and does not contain two adjacent zeros.

%I A247648 #53 Jan 05 2025 19:51:40
%S A247648 1,3,5,7,11,13,15,21,23,27,29,31,43,45,47,53,55,59,61,63,85,87,91,93,
%T A247648 95,107,109,111,117,119,123,125,127,171,173,175,181,183,187,189,191,
%U A247648 213,215,219,221,223,235,237,239,245,247,251,253
%N A247648 Numbers whose binary expansion begins and ends with 1 and does not contain two adjacent zeros.
%C A247648 Decimal equivalents of A247647.
%C A247648 A265716(a(n)) = A265705(2*a(n),a(n)) = 2*a(n). - _Reinhard Zumkeller_, Dec 15 2015
%C A247648 The viabin numbers of the integer partitions having distinct parts (for the definition of viabin number see comment in A290253). For example, 109 is in the sequence because it is the viabin number of the integer partition [5,4,2]; 121 is not in the sequence because it is the viabin number of the integer partition [5,4,4]. - _Emeric Deutsch_, Aug 29 2017
%H A247648 Reinhard Zumkeller, <a href="/A247648/b247648.txt">Table of n, a(n) for n = 1..121392</a> (all terms < 2^24; first 1000 terms from Chai Wah Wu)
%H A247648 Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/57-5/Hinz-Stockmeyer.pdf">Discovering Fibonacci Numbers, Fibonacci Words, and a Fibonacci Fractal in the Tower of Hanoi</a>, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 72-83.
%H A247648 Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Hinz/hinz5.html">Precious Metal Sequences and Sierpinski-Type Graphs</a>, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
%H A247648 N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.
%H A247648 Stefan Witzel, <a href="https://doi.org/10.1007/s10711-017-0247-8">On panel-regular ~A2 lattices</a>, Geom. Dedicata 191, 85-135 (2017).
%e A247648 109 is in the sequence because its binary expansion is 1101101.
%p A247648 vitopart := proc (n) local L, i, j, N, p, t: N := 2*n: L := ListTools:-Reverse(convert(N, base, 2)): j := 0: for i to nops(L) do if L[i] = 0 then j := j+1: p[j] := numboccur(L[1 .. i], 1) end if end do: sort([seq(p[t], t = 1 .. j)], `>=`) end proc: a := proc (n) if n = 1 then 1 elif `mod`(n, 2) = 0 then a((1/2)*n) elif `mod`(n, 2) = 1 and `mod`((1/2)*n-1/2, 2) = 0 then a((1/2)*n-1/2)+1 else a((1/2)*n-1/2) end if end proc: A := {}: for n to 254 do if a(n) = nops(vitopart(n)) then A := `union`(A, {n}) else end if end do: A; # program is based on my comment; the command vitopart(n) yields the integer partition having viabin number n. # _Emeric Deutsch_, Aug 29 2017
%t A247648 Select[Range@ 256, And[First@ # == Last@ # == 1, NoneTrue[Map[Length, Select[Split[#], First@ # == 0 &]], # > 1 &]] &@ IntegerDigits[#, 2] &] (* _Michael De Vlieger_, Aug 29 2017 *)
%o A247648 (Python)
%o A247648 A247648_list = [n for n in range(1,10**5) if n % 2 and not '00' in bin(n)]
%o A247648 # _Chai Wah Wu_, Sep 25 2014
%o A247648 (Haskell)
%o A247648 import Data.Set (singleton, deleteFindMin, insert)
%o A247648 a247648 n = a247648_list !! (n-1)
%o A247648 a247648_list = f $ singleton 1 where
%o A247648    f s = x : f (insert (4 * x + 1) $ insert (2 * x + 1) s')
%o A247648          where (x, s') = deleteFindMin s
%o A247648 -- _Reinhard Zumkeller_, Sep 25 2014
%o A247648 (PARI) isok(k) = if (k%2, my(b=binary(k)); #select(x->(x==0), vector(#b-1, k, b[k]+b[k+1])) == 0); \\ _Michel Marcus_, Jun 15 2024
%Y A247648 Cf. A247647, A247649, A253085.
%Y A247648 Cf. A247875 (complement).
%Y A247648 Cf. A265716, A265705.
%K A247648 nonn,base
%O A247648 1,2
%A A247648 _N. J. A. Sloane_, Sep 25 2014