A102687 -id:A102687 - cp's OEIS frontend

A103948 a(n) is the number of distinct n-th powers of functions {1, 2, 3} -> {1, 2, 3}.

1, 27, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19, 12, 21, 10, 21, 12, 19

Offset: 0

David Wasserman, Feb 21 2005

Index entries for linear recurrences with constant coefficients, signature (-1, 0, 1, 1).

Cf. A102687, A102709, A103947, A103949, A103950.

Row n=3 of A247026.

Join[{1, 27},LinearRecurrence[{-1, 0, 1, 1},{12, 19, 12, 21},68]] (* Ray Chandler, Sep 08 2015 *)

For n > 1, a(n) = a(n+6).

G.f.: (6*x^5-3*x^4-30*x^3-39*x^2-28*x-1) / ((x-1)*(x+1)*(x^2+x+1)). - Colin Barker, Aug 07 2013

A163861 Number of different 5th-power (quintic) mappings of a finite set of n elements into itself.

Original entry on oeis.org

1, 1, 4, 21, 148, 1281, 12942, 150955, 2042272, 31912425, 567737326

Offset: 0

Carlos Alves, Aug 05 2009

The same as A102687 (square composition), A163859 (cubic composition), A163860 (4th-power composition), here a(n) is the number of different mappings g that admit at least one mapping f as the 5th-order root (g=fofofofof) in terms of the composition.

Cf. A102687, A163859, A163860.

Column k=5 of A247026.

a(0) from Alois P. Heinz, Sep 09 2014

a(8)-a(10) from Bert Dobbelaere, Jan 24 2019

A163948 The number of functions on a finite set that are obtainable by a composition power (excluding identity as power).

Original entry on oeis.org

1, 4, 21, 172, 1725, 21066, 307111

Offset: 1

Carlos Alves, Aug 06 2009

The complementary set to A163947.

Cf. A163947, see also A102687, A163859, A163860, A163861.

a(n) = n^n - A163947(n).

A368830 Number of square unlabeled endofunctions from n points to themselves.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 47, 105

Offset: 0

Keith J. Bauer, Jan 07 2024

Also known as square maps or square mapping patterns.

Two endofunctions are taken to be equivalent up to labeling if one is the conjugation of the other by a permutation. (Conjugation is applying the inverse permutation, the endofunction, and then the permutation, in that order. This is equivalent to permuting the "labels" of the set.)

			The a(3) = 4 square endofunctions are:
  1->1, 2->2, 3->3
  1->1, 2->1, 3->1 (equivalent to any constant function)
  1->1, 2->2, 3->1 (equivalent to any function consisting of 2 1-cycles)
  1->2, 2->3, 3->1 (equivalent to any 3-cycle)
Each function listed here is its own square root, except for the 3-cycle, whose square root is its inverse.

Cf. A001372, A102687 (labeled version).

A368867 Number of labeled mappings from n points to themselves with unique square root (endofunctions).

Original entry on oeis.org

1, 1, 2, 2, 56, 544, 5064, 69348, 1210320

Offset: 0

Keith J. Bauer, Jan 08 2024

A mapping f has a unique square root if there exists a unique g such that gg = f.

			For n = 3, the two 3-cycles are unique square roots of each other.
Note that the identity map has more than one square root (i.e., 1->2, 2->1, 3->3 and itself).
Another non-example: 1->1, 2->2, 3->1 has two square roots: itself and 1->2, 2->1, 3->2.
In fact, the only endofunctions on {1,2,3} with unique square roots are the two 3-cycles, so a(3) = 2.

A088994 (permutations only) < This sequence < A102687 (any square maps) < A000312 (all maps).

function increment(size, t)
  t[1] = t[1] + 1
  local index = 1
  while t[index] > size do
    t[index] = 1
    index = index + 1
    if index > size then return true end
    t[index] = t[index] + 1
  end
  return false
end
function get_initial(size)
  local return_value = {}
  for i = 1, size do return_value[i] = 1 end
  return return_value
end
function compute(size)
  candidate = get_initial(size)
  return_value = 0
  repeat
    fun_root = get_initial(size)
    fun_root_count = 0
    repeat
      for i = 1, size do
        if candidate[i] ~= fun_root[fun_root[i]] then
          goto next_fun_root
        end
      end
      fun_root_count = fun_root_count + 1
      if (fun_root_count == 2) then break end
      ::next_fun_root::
    until (increment(size, fun_root))
    if (fun_root_count == 1) then
      return_value = return_value + 1
    end
  until (increment(size, candidate))
  return return_value
end

a(7)-a(8) from Andrew Howroyd, Jan 09 2024

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A103948 a(n) is the number of distinct n-th powers of functions {1, 2, 3} -> {1, 2, 3}.

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A163861 Number of different 5th-power (quintic) mappings of a finite set of n elements into itself.

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A163948 The number of functions on a finite set that are obtainable by a composition power (excluding identity as power).

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A368830 Number of square unlabeled endofunctions from n points to themselves.

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A368867 Number of labeled mappings from n points to themselves with unique square root (endofunctions).

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