A379340 a(n) is the number of pairs of similar integer-sided triangles whose perimeters sum up to n.

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 5, 0, 4, 4, 6, 0, 11, 0, 12, 7, 10, 0, 23, 4, 14, 10, 23, 0, 37, 0, 28, 15, 24, 14, 54, 0, 30, 20, 57, 0, 69, 0, 55, 43, 44, 0, 98, 12, 72, 32, 76, 0, 107, 30, 107, 39, 70, 0, 177, 0, 80, 76, 120, 40, 161, 0, 128, 55, 172, 0, 237

Offset

0,11

Comments

a(n) is the number of solutions to (k + 1)*(x + y + z) = n in positive integers x, y, z and rational k with x <= y <= z and k >= 1.

There is no pair of integer-sided similar triangles whose perimeters sum up to a prime. Proof: If the integer perimeters u = x1 + y1 + z1 and v = x2 + y2 + z2 sum up to a prime p, they are coprimes and the scale factor is k = v/u. Since positive integer x1 < u, x2 = x1*v/u is not an integer. It follows that a(p) = 0.

Links

Felix Huber, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Similar Triangles

Formula

a(2*m) = A379341(2*m) + A069905(m), a(2*m-1) = A379341(2*m-1) for nonnegative integers m.

a(p) = 0 for prime p.

Example

The a(12) = 5 pairs of integer-sided similar triangles whose perimeters sum up to 12 are (1, 1, 1) and (3, 3, 3) with k = 3; (1, 1, 2) and (2, 2, 4) with k = 2; (1, 1, 4) and (1, 1, 4) with k = 1; (1, 2, 3) and (1, 2, 3) with k = 1; (2, 2, 2) and (2, 2, 2) with k = 1.
The a(21) = 7 pairs of integer-sided similar triangles whose perimeters sum up to 21 are (1, 1, 1) and (6, 6, 6) with k = 6; (1, 1, 5) and (2, 2, 10) with k = 2; (1, 2, 4) and (2, 4, 8) with k = 2; (1, 3, 3) and (2, 6, 6) with k = 2; (2, 2, 3) and (4, 4, 6) with k = 2; (2, 2, 2) and (5, 5, 5) with k = 5/2; (3, 3, 3) and (4, 4, 4) with k = 4/3.

Maple

A379340:=proc(n)
   local a, p, q, s;
   a:=0;
   if isprime(n)=false then
      for q to n/6 do
         for s from 3*q to n/2 by q do
            p:=n*q/s-q;
            if is(p, integer) and igcd(p, q)=1 then
               a:=a+round((s/q)^2/12)
            fi	
         od
      od
   fi;
   return a
end proc;
seq(A379340(n), n=0..72);

Crossrefs

Cf. A010814, A069905, A070088, A366398, A370599, A379341.

Sequence in context: A369743 A374941 A369907 * A257013 A104755 A242690

Adjacent sequences: A379337 A379338 A379339 * A379341 A379342 A379344

Keyword

nonn,new

Author

Felix Huber, Dec 24 2024

Status

approved