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Distinct Trapezoidal Decompositions? (Posted on 2025-02-03) Difficulty: 3 of 5
How many distinct trapezoidal decompositions (sum of consecutive positive integers) does n have?

For example, 15 = 7+8 = 4+5+6 = 1+2+3+4+5 has three.

No Solution Yet Submitted by K Sengupta    
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Solution computer-aided solution | Comment 1 of 4
Wolfram Alpha was used to find the solution to

(f+l)*(l-f+1)/2 = n    for l  (that's a lower case L)

so we could find, if it exists (the formula produces an integer), the last term in the sequence (l) given the first (f). The formula was used in the program:

clearvars
sol=[];
for n=1:25
  ct=0; 
  for f=1:n/2 
    l =   (sqrt(4*f^2 - 4*f + 8*n + 1) - 1)/2;
    if abs(l-round(l))<.0000001
      % disp([n f l])
      ct=ct+1;      
    end    
  end
  disp([n ct])
  sol(end+1)=ct;
end
fprintf('%d,',sol)
fprintf('\n')

The findings:


     n    ways
     
     1     0
     2     0
     3     1
     4     0
     5     1
     6     1
     7     1
     8     0
     9     2
    10     1
    11     1
    12     1
    13     1
    14     1
    15     3
    16     0
    17     1
    18     2
    19     1
    20     1
    21     3
    22     1
    23     1
    24     1
    25     2
0,0,1,0,1,1,1,0,2,1,1,1,1,1,3,0,1,2,1,1,3,1,1,1,2,

The last line was for entry into the OEIS, which found A069283, a(n) = -1 + number of odd divisors of n, but which also notes that it is the


Number of nontrivial ways to write n as sum of at least 2 consecutive integers. That is, we are not counting the trivial solution n=n.

And it does turn out that

for n=1:25
  d=divisors(n);
  d=d(mod(d,2)==1);
  disp([n length(d)-1])
end

produces the same list, though it's slower.

  Posted by Charlie on 2025-02-03 08:34:28
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