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A247648
Numbers whose binary expansion begins and ends with 1 and does not contain two adjacent zeros.
17
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
1,2
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
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).
EXAMPLE
109 is in the sequence because its binary expansion is 1101101.
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 *)
PROG
(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
(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
(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
CROSSREFS
Cf. A247875 (complement).
Sequence in context: A292939 A232011 A153183 * A117203 A081118 A363583
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Sep 25 2014
STATUS
approved