OFFSET
1,2
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
LINKS
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).
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
EXAMPLE
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)
MATHEMATICA
a[ n_] := 2*ChebyshevT[n, 2] - 4; (* Michael Somos, Mar 27 2022 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Randall L Rathbun, Mar 26 2022
STATUS
approved