A253085 -id:A253085 - cp's OEIS frontend

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

N. J. A. Sloane, Sep 25 2014

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

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

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).

Cf. A247647, A247649, A253085.
Cf. A247875 (complement).
Cf. A265716, A265705.

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

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

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

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

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

A247649 Number of terms in expansion of f^n mod 2, where f = 1/x^2 + 1/x + 1 + x + x^2 mod 2.

Original entry on oeis.org

1, 5, 5, 7, 5, 17, 7, 19, 5, 25, 17, 19, 7, 31, 19, 25, 5, 25, 25, 35, 17, 61, 19, 71, 7, 35, 31, 41, 19, 71, 25, 77, 5, 25, 25, 35, 25, 85, 35, 95, 17, 85, 61, 71, 19, 91, 71, 77, 7, 35, 35, 49, 31, 107, 41, 121, 19, 95, 71, 85, 25, 113, 77, 103

Offset: 0

N. J. A. Sloane, Sep 25 2014

nonn

This is the number of cells that are ON after n generations in a one-dimensional cellular automaton defined by the odd-neighbor rule where the neighborhood consists of 5 contiguous cells.

a(n) is also the number of odd entries in row n of A035343. - Leon Rische, Feb 02 2023

			The first few generations are:
..........X..........
........XXXXX........
......X.X.X.X.X......
....XX..X.X.X..XX.... (f^3)
..X...X...X...X...X..
XXXX.XXX.XXX.XXX.XXXX
...
f^3 mod 2 = x^6 + x^5 + x^2 + 1/x^2 + 1/x^5 + 1/x^6 + 1 has 7 terms, so a(3) = 7.
From _Omar E. Pol_, Mar 02 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
  1;
  5;
  5, 7;
  5,17, 7,19;
  5,25,17,19, 7,31,19,25;
  5,25,25,35,17,61,19,71, 7,35,31,41,19,71,25,77;
  5,25,25,35,25,85,35,95,17,85,61,71,19,91,71,77,7,35,35,49,31,107,41,121,19, ...
(End)
It follows from the Generalized Run Length Transform result mentioned in the comments that in each row the first quarter of the terms (and no more) are equal to 5 times the beginning of the sequence itself. It cannot be said that the rows converge (in any meaningful sense) to five times the sequence. - _N. J. A. Sloane_, Mar 03 2015

Chai Wah Wu, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Index entries for sequences related to cellular automata

Cf. A071053, A247647, A247648, A253085, A255490.

Partial sums are in A255654.

import sympy
from functools import reduce
from operator import mul
x = sympy.symbols('x')
f = 1/x**2+1/x+1+x+x**2
A247649_list, g = [1], 1
for n in range(1,1001):
    s = [int(d,2) for d in bin(n)[2:].split('00') if d != '']
    g = (g*f).expand(modulus=2)
    if len(s) == 1:
        A247649_list.append(g.subs(x,1))
    else:
        A247649_list.append(reduce(mul,(A247649_list[d] for d in s)))
# Chai Wah Wu, Sep 25 2014

The values of a(n) for n in A247647 (or A247648) determine all the values, as follows. Parse the binary expansion of n into terms from A247647 separated by at least two zeros: m_1 0...0 m_2 0...0 m_3 ... m_r 0...0. Ignore any number (one or more) of trailing zeros. Then a(n) = a(m_1)*a(m_2)*...*a(m_r). For example, n = 37_10 = 100101_2 is parsed into 1.00.101, and so a(37) = a(1)*a(5) = 5*17 = 85. This is a generalization of the Run Length Transform.

A247647 Binary numbers that begin and end with 1 and do not contain two adjacent zeros.

Original entry on oeis.org

1, 11, 101, 111, 1011, 1101, 1111, 10101, 10111, 11011, 11101, 11111, 101011, 101101, 101111, 110101, 110111, 111011, 111101, 111111, 1010101, 1010111, 1011011, 1011101, 1011111, 1101011, 1101101, 1101111, 1110101, 1110111, 1111011, 1111101, 1111111, 10101011, 10101101, 10101111, 10110101, 10110111, 10111011, 10111101

Offset: 1

N. J. A. Sloane, Sep 25 2014

Chai Wah Wu, Table of n, a(n) for n = 1..1000

See A247648 for the decimal equivalents.

Cf. A007088, A247649, A253085.

a247647 = a007088 . a247648  -- Reinhard Zumkeller, Sep 25 2014

With[{upto=500},Map[FromDigits,Select[IntegerString[Range[1,upto,2],2],StringFreeQ[#,"00"]&]]] (* Paolo Xausa, Dec 06 2023 *)

A247647_list = [int(bin(n)[2:]) for n in range(1,10**5) if n % 2 and not '00' in bin(n)]
# Chai Wah Wu, Sep 25 2014

a(n) = A007088(A247648(n)).

A255490 The subsequence A247649(2^n-1).

Original entry on oeis.org

1, 5, 7, 19, 25, 77, 103, 307, 409, 1229, 1639, 4915, 6553, 19661, 26215, 78643, 104857, 314573, 419431, 1258291, 1677721, 5033165, 6710887, 20132659, 26843545, 80530637, 107374183, 322122547, 429496729, 1288490189, 1717986919, 5153960755

Offset: 0

N. J. A. Sloane, Mar 03 2015

Note that A247649 is not the Run Length Transform of this sequence - compare A253085.

Index entries for linear recurrences with constant coefficients, signature (0,3,0,4)

Cf. A247649, A253085.

LinearRecurrence[{0,3,0,4}, {1,5,7,19}, 40] (* Georg Fischer, Aug 18 2021 *)

G.f.: (1+5*x+4*x^2+4*x^3)/(1-3*x^2-4*x^4).

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

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A247649 Number of terms in expansion of f^n mod 2, where f = 1/x^2 + 1/x + 1 + x + x^2 mod 2.

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A247647 Binary numbers that begin and end with 1 and do not contain two adjacent zeros.

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A255490 The subsequence A247649(2^n-1).

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