A349948 Tetrahedral-sided isosceles Heron triangle pairs.

0, 10, 48, 190, 720, 2698, 10080, 37630, 140448, 524170, 1956240, 7300798, 27246960, 101687050, 379501248, 1416317950, 5285770560, 19726764298, 73621286640, 274758382270, 1025412242448, 3826890587530, 14282150107680, 53301709843198, 198924689265120

Offset: 1

Author

Randall L Rathbun, Mar 26 2022

Keywords

• nonn
• easy

Comments

Isosceles Heron triangle pairs with tetrahedral sides: [t(a(n)+1), t(a(n)+1), t(a(n))] and [t(a(n)+6), t(a(n)+5), t(a(n)+5)] where t(n) = A000292(n) is a tetrahedral number, i.e., t(n) = n*(n+1)*(n+2)/6. The Heron triangle pair areas have been checked for rationality to 100 terms of {a(n)}.

Not all isosceles Heron triangles with tetrahedral sides are generated by this sequence. For example, [t(63),t(50),t(50)] is not included. Also, scalene Heron triangles with tetrahedral sides are not included. For example, [t(111),t(104),t(62)]. - Michael Somos, Mar 27 2022

Area of triangles: T1(n) = (b(n)-2)^2*(b(n)-3)^2*(b(n)-4)*c(n)/48 and T2(n) = (b(n)+2)^2*(b(n)+3)^2*(b(n)+4)*c(n)/48, where b(n) = A003500(n) and c(n) = A052530(n). - Randall L Rathbun, Apr 01 2022

Conjecture: for k a positive integer, the sequence {a(k^n): n >= 1} is a strong divisibility sequence; that is, for n, m >= 1, gcd(a(k^n), a(k^m)) = a(k^gcd(n,m)). - Peter Bala, Dec 03 2022

Examples

			10 is a term, so there exists one Heron isosceles triangle whose sides are the 10th, 11th, and 11th tetrahedral numbers (220, 286, 286) and another whose sides are the 15th, 15th, and 16th tetrahedral numbers (680, 680, 816). Those two triangles have areas 29040 and 221952, respectively. (See the n=2 row of the table below.)
.
             Triangle sides               Triangle sides
     k=    ------------------          --------------------
  n a(n)   T(k) T(k+1) T(k+1)  Area    T(k+5) T(k+5) T(k+6)   Area
  - ----   ---- ------ ------ ------   ------ ------ ------  ------
  1    0      0      1      1      0*      35     35     56     588
  2   10    220    286    286  29040      680    680    816  221952
*(degenerate triangle)
		

Links

• Eric Weisstein's World of Mathematics, Tetrahedral Number
• Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Crossrefs

Cf. A000292, A001075, A003500, A052530.

Programs

Mathematica

a[ n_] := 2*ChebyshevT[n, 2] - 4; (* Michael Somos, Mar 27 2022 *)

PARI

Vec(2*x^2*(5 - x)/(1 - 5*x + 5*x^2 - x^3) + O(x^42))

PARI

{a(n) = 2*polchebyshev(n,1, 2) - 4}; /* Michael Somos, Mar 27 2022 */

Formula

a(n+2) = 4*a(n+1) - a(n) + 8.

From Stefano Spezia, Mar 26 2022: (Start)

G.f.: 2*x^2*(5 - x)/((1-x)*(1 - 4*x +x^2)).

a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for n > 3.

a(n) = (2 + sqrt(3))^n + (2 - sqrt(3))^n - 4. (End)

a(n) = 2*A001075(n) - 4. - Michael Somos, Mar 27 2022