A088452 -id:A088452 - cp's OEIS frontend

A088442 A linear version of the Josephus problem.

1, 3, 1, 3, 9, 11, 9, 11, 1, 3, 1, 3, 9, 11, 9, 11, 33, 35, 33, 35, 41, 43, 41, 43, 33, 35, 33, 35, 41, 43, 41, 43, 1, 3, 1, 3, 9, 11, 9, 11, 1, 3, 1, 3, 9, 11, 9, 11, 33, 35, 33, 35, 41, 43, 41, 43, 33, 35, 33, 35, 41, 43, 41, 43, 129, 131, 129, 131, 137, 139, 137, 139, 129, 131

Offset: 0

N. J. A. Sloane, Nov 09 2003

Or a(n) is in A145812 such that (2*n + 3 - a(n))/2 is in A145812 as well. Note also that a(n) + 2*A090569(n+1) = 2*n + 3. - Vladimir Shevelev, Oct 20 2008

			If n=4, 2n+1 = 9 = 1 + 0*2 + 0*2^2 + 1*2^3, so a(4) = 1 + 0*2 + 1*2^3 = 9.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
C. Groer, The Mathematics of Survival: From Antiquity to the Playground, Amer. Math. Monthly, 110 (No. 9, 2003), 812-825. (This is the sequence W(2n+1).)
Index entries for sequences related to the Josephus Problem

Cf. A006257, A063694, A088443, A088452, A090569, A145812.

a088442 = (+ 1) . a004514  -- Reinhard Zumkeller, Sep 26 2015

function A063694(n)
  if n le 1 then return n;
  else return 4*A063694(Floor(n/4)) + (n mod 2);
  end if; return A063694;
end function;
A088442:= func< n | 2*A063694(n) + 1 >;
[A088442(n): n in [0..100]]; // G. C. Greubel, Dec 05 2022

a:=proc(n) local b: b:=convert(2*n+1,base,2): 1+sum(b[2*j]*2^(2*j-1),j=1..nops(b)/2) end: seq(a(n),n=0..100);
with(Bits): seq(And(2*n+1, convert("aaaaaa", decimal, hex)) + 1, n=0..127); # Georg Fischer, Dec 03 2022

A004514[n_]:= A004514[n]= If[n==0, 0, 2*(n-A004514[Floor[n/2]])];
A088442[n_] := A004514[n] +1;
Table[A088442[n], {n,0,100}] (* G. C. Greubel, Dec 05 2022 *)

def A088442(n): return ((n&((1<<(m:=n.bit_length())+(m&1))-1)//3)<<1)+1 # Chai Wah Wu, Jan 30 2023

def A063694(n):
    if (n<2): return n
    else: return 4*A063694(floor(n/4)) + (n%2)
def A088442(n): return 2*A063694(n) + 1
[A088442(n) for n in range(101)] # G. C. Greubel, Dec 05 2022

To get a(n), write 2n+1 as Sum b_j 2^j, then a(n) = 1 + Sum_{j odd} b_j 2^j.
Equals A004514(n) + 1. - Chris Groer (cgroer(AT)math.uga.edu), Nov 10 2003
a(n) = 2*A063694(n) + 1. - G. C. Greubel, Dec 05 2022

More terms from Emeric Deutsch, May 27 2004

A090569 The survivor w(n,2) in a modified Josephus problem, with a step of 2.

Original entry on oeis.org

1, 1, 3, 3, 1, 1, 3, 3, 9, 9, 11, 11, 9, 9, 11, 11, 1, 1, 3, 3, 1, 1, 3, 3, 9, 9, 11, 11, 9, 9, 11, 11, 33, 33, 35, 35, 33, 33, 35, 35, 41, 41, 43, 43, 41, 41, 43, 43, 33, 33, 35, 35, 33, 33, 35, 35, 41, 41, 43, 43, 41, 41, 43, 43, 1, 1, 3, 3, 1, 1, 3, 3, 9, 9, 11, 11, 9, 9, 11, 11, 1, 1

Offset: 1

John W. Layman, Dec 02 2003

nonn

Arrange n persons {1,2,...,n} consecutively on a line rather than around in a circle. Beginning at the left end of the line, we remove every q-th person until we reach the end of the line. At this point we immediately reverse directions, taking care not to "double count" the person at the end of the line and continue to eliminate every q-th person, but now moving right to left. We continue removing people in this back-and-forth manner until there remains a lone survivor w(n,q).

Or a(n) is in A145812 such that 2n+1-2a(n) is in A145812 as well. Note also that 2a(n)+A088442(n-1)=2n+1. - Vladimir Shevelev, Oct 20 2008

			a(2)=11, since people are eliminated in the order 2, 4, 6, 8, 10, 12, 9, 5, 1, 7, 3, leaving 11 as the survivor.

Chris Groër, The Mathematics of Survival: From Antiquity to the Playground, Amer. Math. Monthly, 110 (No. 9, 2003), 812-825.
Index entries for sequences related to the Josephus Problem

Cf. A088443, A088452.

def A090569(n): return (n-1&((1<<(m:=(n-1).bit_length())+(m&1^1))-1)//3)+1 # Chai Wah Wu, Jan 30 2023

w(n, 2) = 1 + Sum_{odd j=1..k} b(j)*(2^j), where Sum_{j=0..k} b(j)*(2^j) is the binary expansion of either n or n-1, whichever is odd.

a(n) = A063695(n-1) + 1.

A088443 A linear version of the Josephus problem: a(n) = the function w_3(n).

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 7, 8, 8, 1, 2, 1, 2, 5, 14, 14, 17, 17, 17, 17, 14, 2, 1, 4, 1, 1, 2, 4, 4, 5, 11, 32, 31, 34, 31, 31, 37, 38, 38, 38, 41, 37, 38, 37, 38, 31, 31, 1, 4, 5, 1, 7, 8, 8, 1, 2, 1, 2, 5, 4, 1, 8, 8, 8, 8, 11, 11, 20, 23, 25, 71, 71, 68, 70, 68, 76, 74, 68, 68, 68, 70, 82, 83, 82

Offset: 1

N. J. A. Sloane, Nov 09 2003

nonn

The survivor w(n,3) in a modified Josephus problem, with a step of 3.

See A090569 or the reference for the definition of w(n,q).

Chris Groër, The Mathematics of Survival: From Antiquity to the Playground, Amer. Math. Monthly, 110 (No. 9, 2003), 812-825.
Index entries for sequences related to the Josephus Problem

Cf. A006257, A088442, A088452, A090569.

A recurrence is given in the reference.

Terms computed by Chris Groer (cgroer(AT)math.uga.edu)

More terms from John W. Layman, Feb 05 2004

cp's OEIS Frontend

A090571 Duplicate of A088452.

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A088442 A linear version of the Josephus problem.

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A090569 The survivor w(n,2) in a modified Josephus problem, with a step of 2.

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A088443 A linear version of the Josephus problem: a(n) = the function w_3(n).

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