A338671 a(n) is the number of distinct ways of arranging n identical square tiles into two rectangles.
0, 1, 1, 2, 3, 4, 5, 7, 7, 10, 10, 13, 14, 16, 16, 21, 20, 25, 24, 29, 28, 34, 30, 40, 36, 43, 40, 50, 44, 55, 49, 60, 55, 66, 55, 75, 64, 75, 70, 86, 72, 92, 77, 97, 87, 103, 84, 116, 94, 114, 104, 126, 102, 135, 109, 138, 123, 143, 117, 164, 128, 153, 138, 171
Offset: 1
Keywords
Examples
a(5) = 3, since there are 3 ways to form 2 rectangles from 5 identical square tiles: 1) 2 X 2 and 1 X 1 2) 3 X 1 and 2 X 1 3) 4 X 1 and 1 X 1 Note that rotation through 90 degrees and/or exchanging the order of the two rectangles in a pair naturally do not create a distinct pair. For example, the pair 2 X 1 and 1 X 3 is not distinct from pair 2 above.
Crossrefs
Programs
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PARI
a(n) = {(sum(k=1, n-1, ((numdiv(k)+1)\2)*((numdiv(n-k)+1)\2)) + if(n%2==0, (numdiv(n/2)+1)\2))/2} \\ Andrew Howroyd, Apr 29 2021 -
Python
import numpy as np # This sets the number of terms: nits = 20 # This list will contain the sequence seq = [] # The indices of the sequence: for i in range(1,nits + 1): # This variable counts the pairs found for each total area i count = 0 # The longest side of either rectangle: for a in range(1,i): # The other side of the same rectangle: for b in np.arange(1,1 + min(a,np.floor(i/a))): # Calculate the area of this rectangle and the remaining area: area1 = a*b rem_area = i - a*b # The longer side of the second rectangle: for c in np.arange(1,1 + min(a,rem_area)): # The shorter side of the second rectangle: d = rem_area / c # Check that the solution is valid and not double counted: if d != int(d) or d > min(a,c) or (a == c and d > b): continue # Count the new pair found: count += 1 # Add to the sequence: seq.append(count) for an in seq: print(an)
Formula
G.f.: (B(x)^2 + B(x^2))/2 where B(x) is the g.f. of A038548. - Andrew Howroyd, Apr 29 2021
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