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A226351 Number of ways to tile a fixed 3 X n square grid with 1 X 1, 2 X 2, and 1 X 2 tiles.

%I A226351 #23 Nov 11 2024 22:29:01
%S A226351 1,3,26,163,1125,7546,51055,344525,2326760,15709977,106079739,
%T A226351 716273960,4836475953,32657123299,220509407586,1488936665619,
%U A226351 10053686907525,67885102598386,458377829683919,3095086053853821,20898824215523616
%N A226351 Number of ways to tile a fixed 3 X n square grid with 1 X 1, 2 X 2, and 1 X 2 tiles.
%H A226351 Andrew Woods, <a href="/A226351/b226351.txt">Table of n, a(n) for n = 0..100</a>
%H A226351 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,19,1,-26,1,6).
%F A226351 Recurrence: a(n) = 4*a(n-1)+19*a(n-2)+a(n-3)-26*a(n-4)+a(n-5)+6*a(n-6) for n>5, a(0)=1, a(1)=3, a(2)=26, a(3)=163, a(4)=1125, a(5)=7546.
%F A226351 G.f.: (1-x-5*x^2+x^3+2*x^4)/(1-4*x-19*x^2-x^3+26*x^4-x^5-6*x^6).
%t A226351 LinearRecurrence[{4, 19, 1, -26, 1, 6}, {1, 3, 26, 163, 1125, 7546}, 21] (* _T. D. Noe_, Jun 04 2013 *)
%o A226351 (Python)
%o A226351 # Depth-first search on 3 rows and n columns
%o A226351 # Produces "count" and the list "result[]"
%o A226351 # Omit the 2nd-last line if memory runs low
%o A226351 n=5; rows=3
%o A226351 count=0; result=[]
%o A226351 def f(b, row=0, col=-1):
%o A226351   global count
%o A226351   for i in range(row, len(b)):
%o A226351     for j in range((col+1 if i==row else 0), len(b[0])):
%o A226351       if b[i][j]==' ':
%o A226351         if i<len(b)-1:
%o A226351           if b[i+1][j]==' ':
%o A226351             f(b[:i]+[b[i][:j]+'^'+b[i][j+1:], b[i+1][:j]+'V'+b[i+1][j+1:]]+b[i+2:], i, j)
%o A226351             if j<len(b[0])-1:
%o A226351               if b[i][j+1]==' ' and b[i+1][j:j+2]==' ':
%o A226351                 f(b[:i]+[b[i][:j]+'/\\'+b[i][j+2:], b[i+1][:j]+'\\/'+b[i+1][j+2:]]+b[i+2:], i, j)
%o A226351         if j<len(b[0])-1:
%o A226351           if b[i][j+1]==' ':
%o A226351             f(b[:i]+[b[i][:j]+'<>'+b[i][j+2:]]+b[i+1:], i, j)
%o A226351   count+=1
%o A226351   result.append(b) # omit this line
%o A226351 f([' '*n]*rows); print(count)
%Y A226351 Cf. A226348.
%K A226351 nonn,easy
%O A226351 0,2
%A A226351 _Andrew Woods_, Jun 04 2013