cp's OEIS Frontend

A287898 Number of ways to go up and down n stairs, with fewer than 4 stairs at a time, stepping on each stair at least once.

%I A287898 #56 Jun 03 2017 07:27:42
%S A287898 1,3,9,27,79,233,687,2025,5969,17595,51865,152883,450655,1328401,
%T A287898 3915743,11542481,34023905,100292659,295633833,871443275,2568763439,
%U A287898 7571973753,22319994767,65792907193,193938514865,571674807403,1685132453689,4967284459107
%N A287898 Number of ways to go up and down n stairs, with fewer than 4 stairs at a time, stepping on each stair at least once.
%C A287898 Also the number of words using 0, 1 and 2 which have n-1 length and don't appear 0000 or 1111.
%H A287898 Colin Barker, <a href="/A287898/b287898.txt">Table of n, a(n) for n = 1..1000</a>
%H A287898 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,1).
%F A287898 a(n+4) = 2*a(n+3) + 2*a(n+2) + 2*a(n+1) + a(n).
%F A287898 G.f.: x*(1 + x)*(1 + x^2) / (1 - 2*x - 2*x^2 - 2*x^3 - x^4). - _Colin Barker_, Jun 02 2017
%e A287898 n = 2
%e A287898 0->1->2->0 (0), 0->2->1->0 (1), 0->1->2->1->0 (2). So a(2) = 3.
%e A287898 n = 3
%e A287898 0->1->2->3->0    (00), 0->1->3->2->0    (01), 0->1->2->3->2->0    (02),
%e A287898 0->2->3->1->0    (10), 0->3->2->1->0    (11), 0->2->3->2->1->0    (12),
%e A287898 0->1->2->3->1->0 (20), 0->1->3->2->1->0 (21), 0->1->2->3->2->1->0 (22). So a(3) = 9.
%e A287898 ...
%e A287898 n = 5
%e A287898 0->1->2->3->5->4->0 (0001), ... , 0->4->5->3->2->1->0 (1110),
%e A287898 0->4->5->4->3->2->1->0 (1112), ... , 0->1->2->3->4->5->4->3->2->1->0 (2222).
%e A287898 So a(5) = 81 - 2 = 79.
%t A287898 CoefficientList[Series[(1 + x)*(1 + x^2)/(1 - 2*x - 2*x^2 - 2*x^3 - x^4), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Jun 02 2017 *)
%o A287898 (Ruby)
%o A287898 def f(ary, n)
%o A287898   return false if ary.size < n
%o A287898   a = ary[-1]
%o A287898   ary[-n..-2].all?{|i| i == a}
%o A287898 end
%o A287898 def A(k, n)
%o A287898   f_ary = [[]]
%o A287898   ary = [1]
%o A287898   (n - 1).times{
%o A287898     b_ary = []
%o A287898     f_ary.each{|i|
%o A287898       i0, i1, i2 = i + [0], i + [1], i + [2]
%o A287898       b_ary << i0 if !f(i0, k)
%o A287898       b_ary << i1 if !f(i1, k)
%o A287898       b_ary << i2
%o A287898     }
%o A287898     f_ary = b_ary
%o A287898     ary << f_ary.size
%o A287898   }
%o A287898   ary
%o A287898 end
%o A287898 p A(4, 10)
%o A287898 (PARI) Vec(x*(1 + x)*(1 + x^2) / (1 - 2*x - 2*x^2 - 2*x^3 - x^4) + O(x^30)) \\ _Colin Barker_, Jun 02 2017
%Y A287898 Cf. A001333, A233828(n-1).
%K A287898 nonn,easy
%O A287898 1,2
%A A287898 _Seiichi Manyama_, Jun 02 2017
%E A287898 More terms from _Colin Barker_, Jun 02 2017