A292793 1/4 of the number of self-avoiding paths that are made of alternated vertical and horizontal n consecutive steps. (start point is different from end point.)

1, 2, 4, 8, 16, 29, 54, 98, 176, 318, 572, 1026, 1826, 3255, 5794, 10233, 18172, 32012, 56488, 99469, 175034, 307479, 540068, 947235, 1659907, 2908958, 5095019, 8917677, 15598100, 27281252, 47718310, 83298748, 145405769, 253641303, 442352671, 770769569, 1343166519, 2339093953

Offset: 2

Author

Seiichi Manyama, Sep 23 2017

Keywords

• nonn

Examples

			a(2) = 1;
   E
   |
   *
   |
S--*
a(3) = 2;
   *--*--*--E     E--*--*--*
   |                       |
   *                       *
   |                       |
S--*                    S--*
a(4) = 4;
            E                                      E
            |                                      |
            *                                      *
            |                                      |
            *                                      *
            |                                      |
            *                                      *
            |                                      |
   *--*--*--*        *--*--*--*     *--*--*--*     *--*--*--*
   |                 |        |     |        |              |
   *                 *        *     *        *              *
   |                 |        |     |        |              |
S--*              S--*        *     *     S--*           S--*
                              |     |
                              E     E
		

Crossrefs

Cf. A101856.

Programs

Ruby

def A292793(n)
  ary = [1]
  b_ary = [[[0, 0], [1, 0], [1, 1], [1, 2]]]
  s = 4
  (3..n).each{|i|
    s += i
    f_ary, b_ary = b_ary, []
    if i % 2 == 1
      f_ary.each{|a|
        b = a.clone
        x, y = *b[-1]
        b += (1..i).map{|j| [x + j, y]}
        b_ary << b if b.uniq.size == s
        c = a.clone
        x, y = *c[-1]
        c += (1..i).map{|j| [x - j, y]}
        b_ary << c if c.uniq.size == s
      }
    else
      f_ary.each{|a|
        b = a.clone
        x, y = *b[-1]
        b += (1..i).map{|j| [x, y + j]}
        b_ary << b if b.uniq.size == s
        c = a.clone
        x, y = *c[-1]
        c += (1..i).map{|j| [x, y - j]}
        b_ary << c if c.uniq.size == s
      }
    end
    ary << b_ary.size
  }
  ary
end
p A292793(16)

Extensions

a(25)-a(39) from Bert Dobbelaere, Sep 14 2019