A382268 Numbers k such that a right triangle can be formed from a chain of linked rods of lengths 1, 2, 3, ..., k, with the perimeter equal to the total length.

15, 20, 24, 35, 39, 44, 48, 55, 56, 63, 75, 76, 80, 84, 91, 95, 99, 104, 111, 119, 120, 132, 135, 140, 143, 144, 152, 155, 168, 175, 176, 187, 188, 195, 203, 207, 215, 216, 219, 224, 252, 259, 260, 264, 272, 275, 279, 287, 288, 296, 299, 308, 315, 320, 324, 335, 351, 360

Offset

1,1

Comments

The corresponding perimeters T(a(n)) must be in the intersection of T = A000217 (triangular numbers) with A010814 (perimeters of integer sided right triangles). A number k is in the sequence if there exists a solution {k1, k2, k3} with k > k1 > k2 > k3 >= 0 such that for a < b < c in { T(k1) - T(k2), T(k2) - T(k3), T(k) - T(k1) + T(k3) } one has a^2 + b^2 = c^2. - M. F. Hasler, Mar 20 2025

Links

Daniel Mondot, Table of n, a(n) for n = 1..3052

Example

The first triangle is when k = 15. The segments are [6+7+8+9+10] [11+12+13+14] [15+1+2+3+4+5]. The sums of the segments are (40, 50, 30), which is 10 times the primitive right triangle (3, 4, 5).
The second term, k = 20, corresponds to 5 distinct solutions:
  S1 = {18, 16, 9}: a = 9+...+1 + 20+19 = 84, b = 18+17 = 35, c = 16+...+10 = 91,
  S2 = {17, 11, 3}: a = 20+19+18 + 3+2+1 = 63, c = 17+...+12 = 87, b = 11+...+4 = 60,
  S3 = {17, 11, 2}: a = 20+19+18 + 2+1 = 60, c = 17+...+12 = 87, b = 11+...+3 = 63,
  S4 = {16, 9, 4}: a = 20+...+17 + 4+...+1 = 84, c = 16+...+10 = 91, b = 9+...+5 = 35,
  S5 = {15, 8, 1}: c = 20+...+16 + 1 = 91, a = 15+...+9 = 84, b = 8+...+2 = 35.
We note that S2 and S3, and S1, S4 and S5, have the same side lengths, but different decompositions.

Prog

(PARI) select( {is_A382268(n)=my(Tn=n*(n+1)\2, T1, T2, S); Tn%2==0 && is_A005279(Tn\2) && forstep(n1=n-1, sqrtint(Tn-1)+1, -1, T1=n1*(n1+1)\2; forstep(n2=n1-1, sqrtint(2*T1-Tn-1)+1, -1, T2=n2*(n2+1)\2; forstep(n3=n2-1, 0, -1, #(S=Set([Tn-T1+S=n3*(n3+1)\2, T2-S, T1-T2]))>2 && S[3]^2 == S[1]^2+S[2]^2 && return(S))))}, [1..100])\\ M. F. Hasler, Mar 22 2025

Crossrefs

Cf. A000217, A010814, A380867, A380868, A380875.

Sequence in context: A102495 A133288 A322710 * A316743 A014603 A070222

Adjacent sequences: A382265 A382266 A382267 * A382269 A382270 A382271

Keyword

nonn

Author

Ali Sada and Daniel Mondot, Mar 19 2025

Status

approved