The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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