This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349948 #75 Jan 25 2023 10:08:29
%S A349948 0,10,48,190,720,2698,10080,37630,140448,524170,1956240,7300798,
%T A349948 27246960,101687050,379501248,1416317950,5285770560,19726764298,
%U A349948 73621286640,274758382270,1025412242448,3826890587530,14282150107680,53301709843198,198924689265120
%N A349948 Tetrahedral-sided isosceles Heron triangle pairs.
%C A349948 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)}.
%C A349948 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
%C A349948 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
%C A349948 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
%H A349948 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>
%H A349948 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).
%F A349948 a(n+2) = 4*a(n+1) - a(n) + 8.
%F A349948 From _Stefano Spezia_, Mar 26 2022: (Start)
%F A349948 G.f.: 2*x^2*(5 - x)/((1-x)*(1 - 4*x +x^2)).
%F A349948 a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for n > 3.
%F A349948 a(n) = (2 + sqrt(3))^n + (2 - sqrt(3))^n - 4. (End)
%F A349948 a(n) = 2*A001075(n) - 4. - _Michael Somos_, Mar 27 2022
%e A349948 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.)
%e A349948 .
%e A349948 Triangle sides Triangle sides
%e A349948 k= ------------------ --------------------
%e A349948 n a(n) T(k) T(k+1) T(k+1) Area T(k+5) T(k+5) T(k+6) Area
%e A349948 - ---- ---- ------ ------ ------ ------ ------ ------ ------
%e A349948 1 0 0 1 1 0* 35 35 56 588
%e A349948 2 10 220 286 286 29040 680 680 816 221952
%e A349948 *(degenerate triangle)
%t A349948 a[ n_] := 2*ChebyshevT[n, 2] - 4; (* _Michael Somos_, Mar 27 2022 *)
%o A349948 (PARI) Vec(2*x^2*(5 - x)/(1 - 5*x + 5*x^2 - x^3) + O(x^42))
%o A349948 (PARI) {a(n) = 2*polchebyshev(n,1, 2) - 4}; /* _Michael Somos_, Mar 27 2022 */
%Y A349948 Cf. A000292, A001075, A003500, A052530.
%K A349948 nonn,easy
%O A349948 1,2
%A A349948 _Randall L Rathbun_, Mar 26 2022