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A349948 Tetrahedral-sided isosceles Heron triangle pairs. 0
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 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=1..25.

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 - 5*x + 5*x^2 - x^3).

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

Cf. A000292, A001075, A003500, A052530.

Sequence in context: A271638 A238916 A084857 * A330170 A126734 A264266

Adjacent sequences:  A349945 A349946 A349947 * A349949 A349950 A349951

KEYWORD

nonn,easy

AUTHOR

Randall L Rathbun, Mar 26 2022

STATUS

approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)