cp's OEIS Frontend

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.

Previous Showing 31-32 of 32 results.

A349948 Tetrahedral-sided isosceles Heron triangle pairs.

Original entry on oeis.org

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

Views

Author

Randall L Rathbun, Mar 26 2022

Keywords

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

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

A378908 Square array, read by descending antidiagonals, where each row n comprises the integers w >= 1 such that A000037(n)*w^2+4 is a square.

Original entry on oeis.org

4, 24, 2, 140, 8, 1, 816, 30, 3, 4, 4756, 112, 8, 40, 6, 27720, 418, 21, 396, 96, 2, 161564, 1560, 55, 3920, 1530, 12, 12, 941664, 5822, 144, 38804, 24384, 70, 456, 6, 5488420, 21728, 377, 384120, 388614, 408, 17316, 120, 1, 31988856, 81090, 987, 3802396
Offset: 1

Views

Author

Charles L. Hohn, Dec 10 2024

Keywords

Comments

Also, integers w >= 1 for each row n >= 1 such that z+(1/z) is an integer, where x = A000037(n), y = w*sqrt(x), and z = (y+ceiling(y))/2.
All terms of row n are positive integer multiples of T(n, 1).
Limit_{k->oo} T(n, k+1)/T(n, k) = (sqrt(b^2-4)+b)/2 where b=T(n, 2)/T(n, 1).

Examples

			n=row index; x=nonsquare integer of index n (A000037(n)):
 n  x    T(n, k)
------+---------------------------------------------------------------------
 1  2 |  4,   24,   140,     816,      4756,       27720,        161564, ...
 2  3 |  2,    8,    30,     112,       418,        1560,          5822, ...
 3  5 |  1,    3,     8,      21,        55,         144,           377, ...
 4  6 |  4,   40,   396,    3920,     38804,      384120,       3802396, ...
 5  7 |  6,   96,  1530,   24384,    388614,     6193440,      98706426, ...
 6  8 |  2,   12,    70,     408,      2378,       13860,         80782, ...
 7 10 | 12,  456, 17316,  657552,  24969660,   948189528,   36006232404, ...
 8 11 |  6,  120,  2394,   47760,    952806,    19008360,     379214394, ...
 9 12 |  1,    4,    15,      56,       209,         780,          2911, ...
10 13 |  3,   33,   360,    3927,     42837,      467280,       5097243, ...
11 14 |  8,  240,  7192,  215520,   6458408,   193536720,    5799643192, ...
12 15 |  2,   16,   126,     992,      7810,       61488,        484094, ...
13 17 | 16, 1056, 69680, 4597824, 303386704, 20018924640, 1320945639536, ...
14 18 |  8,  272,  9240,  313888,  10662952,   362226480,   12305037368, ...
...

Crossrefs

Rows from n = 1 (nonzero terms): A005319, A052530, A001906, A122652, A001080*2, A001542, A239365, A075844, A001353, A075835, A068204*2, A001090*2, A121740*2, A202299*2.
Cf. A000037, A002420, A033999, A048942, A180495, A298675.

Programs

  • PARI
    row(n)={my(v=List()); for(t=3, oo, if((t^2-4)%x>0 || !issquare((t^2-4)/x), next); listput(v, sqrtint((t^2-4)/x)); break); listput(v, v[1]*sqrtint(v[1]^2*x+4)); while(#v<10, listput(v, v[#v]*(v[2]/v[1])-v[#v-1])); Vec(v)}
for(n=1, 20, x=n+floor(1/2+sqrt(n)); print (n, " ", x, " ", row(n)))

Formula

For x = A000037(n) (nonsquare integer of index n):
If x is not the sum of 2 squares, then T(n, 1) = A048942(n); otherwise, T(n, 1) is a positive integer multiple of A048942(n).
For j in {-2, 1, 2, 4}, if x-j is a square (except 2-2=0^2 or 5-1=2^2), then T(n, 1) = (4/abs(j))*sqrt(x-j) and T(n, 2) = T(n, 1)^3/(4/abs(j)) + sign(j)*2*T(n, 1).
For j in {1, 4}, if x+j is a square, then T(n, 1) = 2/sqrt(4/j) and T(n, 2) = (4/j)*sqrt(x+j).
For k >= 2, T(n, k) = T(n, k-1)*sqrt(T(n, 1)^2*x+4) - [k>=3]*T(n, k-2).
T(n, 2) = Sum_{i=0..oo}(T(n, 1)^(2-2*i) * x^((1-2*i)/2) * A002420(i) * A033999(i)).
If T(n, 1) is even, then T(n, 2) = T(n, 1)*A180495(n); if T(n, 1) is odd and x is even, then T(n, 2) = T(n, 1)*sqrt(A180495(n)+2); if T(n, 1) and x are both odd, then T(n, 2) is a factor of T(n, 1)*A180495(n).
For k >= 3, T(n, k) = T(n, k-1)*(T(n, 2)/T(n, 1)) - T(n, k-2) = T(n, 1)*A298675(T(n, 2)/T(n, 1), k-1) + T(n, k-2) = sqrt((A298675(T(n, 2)/T(n, 1), k)^2-4)/x).
Previous Showing 31-32 of 32 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.