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.

Showing 1-2 of 2 results.

A357276 Middle side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3 = 120 degrees.

Original entry on oeis.org

5, 8, 16, 24, 33, 35, 39, 56, 45, 63, 51, 85, 80, 57, 77, 95, 120, 120, 88, 91, 115, 143, 112, 161, 105, 175, 165, 195, 208, 160, 168, 145, 224, 203, 187, 221, 155, 261, 217, 192, 279, 209, 288, 247, 320, 272, 323, 280, 231, 315, 273, 259, 385, 357, 333, 304, 399, 352, 253, 407, 299, 287, 440
Offset: 1

Views

Author

Bernard Schott, Sep 25 2022

Keywords

Comments

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of largest side c, and if largest sides coincide, then by increasing order of the smallest side. This sequence lists the b's.
For the corresponding primitive triples and miscellaneous properties and references, see A357274.
Solutions b of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
Also, b is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with b = 2*u*v + v^2.
This sequence is not increasing. For example, a(8) = 56 for triangle with largest side c = 61 while a(9) = 45 for triangle with largest side c = 67.
Differs from A088586, the first 20 terms are the same then a(21) = 115 while A088586(21) = 143.
A229849 gives all the possible values of the middle side b, in increasing order without repetition, for primitive triples, while A229859 gives all the possible values of the middle side b, in increasing order without repetition, but for all triples, not necessarily primitive.

Examples

			a(17) = a(18) = 120 since 17th and 18th triples are respectively (13, 120, 127) and (23, 120, 133).

Crossrefs

Cf. A357274 (triples), A357275 (smallest side), this sequence (middle side), A357277 (largest side), A357278 (perimeter).
Cf. also A088586, A229849, A229859.

Programs

  • Maple
    for c from 5 to 500 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a,b)=1 and a

A088514 Smaller section of edge A089025(n) by integral cevian A088513(n).

Original entry on oeis.org

3, 7, 5, 11, 7, 13, 16, 9, 32, 17, 40, 11, 19, 55, 40, 24, 13, 23, 65, 69, 25, 56, 15, 75, 32, 104, 56, 29, 17, 87, 85, 31, 119, 72, 93, 64, 144, 95, 40, 35, 133, 21, 136, 105, 37, 105, 111, 88, 23, 185, 152, 176, 80, 41, 115
Offset: 1

Views

Author

Lekraj Beedassy, Nov 14 2003

Keywords

Comments

Obviously, larger section is A089025(n) - a(n).

Programs

  • Mathematica
    findPrimIntEquiSmallSection[maxC_] :=
Reap[Do[Do[
     With[{cevian = Abs[c E^((2 \[Pi] I)/6) - a]},
      If[FractionalPart[cevian] == 0 && GCD[a, c] == 1,
       Sow[a]; Break[]]], {a, Floor[c/2],
      1, -1}], {c, maxC}]][[2, 1]]
(* Andrew Turner, Aug 04 2017 *)

Formula

Larger section is A089025(n) - a(n) = A088586(n). a(n) = (A089025(n) - A088587(n))/2. - Lekraj Beedassy, Nov 25 2003

Extensions

a(7) changed from 18 to 16 by W. D. Smith, Mar 04 2012 (N. J. A. Sloane, Dec 05 2012)
a(42) changed from 105 to 21, and a(52)=176 inserted by Andrew Turner, Aug 04 2017
Showing 1-2 of 2 results.

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

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