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

1, 3, 5, 7, 11, 13, 15, 21, 23, 27, 29, 31, 43, 45, 47, 53, 55, 59, 61, 63, 85, 87, 91, 93, 95, 107, 109, 111, 117, 119, 123, 125, 127, 171, 173, 175, 181, 183, 187, 189, 191, 213, 215, 219, 221, 223, 235, 237, 239, 245, 247, 251, 253

Offset: 1

Author

N. J. A. Sloane, Sep 25 2014

Keywords

• nonn
• base

Comments

Decimal equivalents of A247647.

A265716(a(n)) = A265705(2*a(n),a(n)) = 2*a(n). - Reinhard Zumkeller, Dec 15 2015

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

Examples

			109 is in the sequence because its binary expansion is 1101101.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..121392 (all terms < 2^24; first 1000 terms from Chai Wah Wu)
• Andreas M. Hinz and Paul K. Stockmeyer, Discovering Fibonacci Numbers, Fibonacci Words, and a Fibonacci Fractal in the Tower of Hanoi, The Fibonacci Quarterly (2019) Vol. 57, No. 5, 72-83.
• Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
• Stefan Witzel, On panel-regular ~A2 lattices, Geom. Dedicata 191, 85-135 (2017).

Crossrefs

Cf. A247647, A247649, A253085.

Cf. A247875 (complement).

Cf. A265716, A265705.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a247648 n = a247648_list !! (n-1)
a247648_list = f $ singleton 1 where
   f s = x : f (insert (4 * x + 1) $ insert (2 * x + 1) s')
         where (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Sep 25 2014

Maple

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

Mathematica

Select[Range@ 256, And[First@ # == Last@ # == 1, NoneTrue[Map[Length, Select[Split[#], First@ # == 0 &]], # > 1 &]] &@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Aug 29 2017 *)

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

Python

A247648_list = [n for n in range(1,10**5) if n % 2 and not '00' in bin(n)]
# Chai Wah Wu, Sep 25 2014