A228693 -id:A228693 - cp's OEIS frontend

A211168 Exponent of alternating group An.

1, 1, 3, 6, 30, 60, 420, 420, 1260, 2520, 27720, 27720, 360360, 360360, 360360, 360360, 6126120, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800, 2329089562800

Offset: 1

Alexander Gruber, Jan 31 2013

a(n) is the smallest natural number m such that g^m = 1 for any g in An.

If m <= n, a m-cycle occurs in some permutation in An if and only if m is odd or m <= n - 2. The exponent is the LCM of the m's satisfying these conditions, leading to the formula below.

			For n = 7, lcm{1,...,5,7} = 420.

Alexander Gruber, Table of n, a(n) for n = 1..2308

Even entries given by the sequence A076100, or the odd entries in the sequence A003418.

The records of this sequence are a subsequence of A002809 and A126098.

for n in [1..40] do
Exponent(AlternatingGroup(n));
end for;

for n in [1..40] do
if n mod 2 eq 0 then
L := [1..n-1];
else
L := Append([1..n-2],n);
end if;
LCM(L);
end for;

Table[If[Mod[n, 2] == 0, LCM @@ Range[n - 1],
  LCM @@ Join[Range[n - 2], {n}]], {n, 1, 100}] (* or *)
a[1] = 1; a[2] = 1; a[3] = 3; a[n_] := a[n] =
  If[Mod[n, 2] == 0, LCM[a[n - 1], n - 2], LCM[a[n - 2], n - 3, n]]; Table[a[n], {n, 1, 40}]

a(n)=lcm(if(n%2,concat([2..n-2],n),[2..n-1])) \\ Charles R Greathouse IV, Mar 02 2014

Explicit:
a(n) = lcm{1, ..., n-1} if n is even.
     = lcm{1, ..., n-2, n} if n is odd.
Recursive:
Let a(1) = a(2) = 1 and a(3) = 3.  Then
a(n) = lcm{a(n-1), n-2} if n is even.
     = lcm{a(n-2), n-3, n} if n is odd.
a(n) = A003418(n)/(1 + [n in A228693]) for n > 1. - Charlie Neder, Apr 25 2019

A343949 Shortest distance from curve start to end along the segments of dragon curve expansion level n, and which is the diameter of the curve as a graph.

Original entry on oeis.org

1, 2, 4, 8, 12, 18, 26, 36, 52, 70, 102, 136, 200, 266, 394, 524, 780, 1038, 1550, 2064, 3088, 4114, 6162, 8212, 12308, 16406, 24598, 32792, 49176, 65562, 98330, 131100, 196636, 262174, 393246, 524320, 786464, 1048610, 1572898, 2097188, 3145764, 4194342, 6291494

Offset: 0

Kevin Ryde, May 05 2021

Expansion level n is the first 2^n segments of the curve, and can be taken as a graph with visited points as vertices and segments as edges.

			Curve n=4:
     *--*  *--*
     |  |  |  |        Start S to end E along segments.
     *--*--*  *--*     Distance a(4) = 12,
        |        |     which is also graph diameter.
  E  *--*     S--*
  |  |
  *--*

Kevin Ryde, Table of n, a(n) for n = 0..1000
Kevin Ryde, Iterations of the Dragon Curve, see index "Diameter".
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-2,2).

Cf. A275970, A083706, A228693.

Cf. A332383, A332384 (curve coordinates).

a(n) = if(n==0,1, my(t=n%2); (3+t)<<(n>>1) + n-4 + t);

a(0) = 1.
a(2*n)   = 3*2^n + 2*n - 4 = 2*A275970(n-1), for n>=1.
a(2*n+1) = 4*2^n + 2*n - 2 = 2*A083706(n).
a(n+1) - a(n) = 2*A228693(n), for n>=1.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 2*a(n-4) + 2*a(n-5) for n >= 6.
G.f.: (1 + x - x^2 + x^3 - 4*x^5) / ((1+x) * (1-x)^2 * (1-2*x^2)).
G.f.: 2 - (1/2)/(1+x) - (9/2)/(1-x) + 1/(1-x)^2  + (3 + 4*x)/(1 - 2*x^2).

cp's OEIS Frontend

A211168 Exponent of alternating group An.

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