A216267 Numbers that are both tetrahedral and pronic.

0, 20, 56, 7140, 1414910

Offset: 1

Author

Alex Ratushnyak, Mar 15 2013

Keywords

• nonn
• fini
• full

Comments

Intersection of A000292 and A002378.

The equation y*(y+1) = x*(x+1)*(x+2)/6 leads to an elliptic curve, which has a finite number of solutions, all of which are already listed. - Max Alekseyev, Dec 28 2024

Crossrefs

Cf. A000292, A002378, A027568, A029549, A003556.

Programs

Mathematica

t = {}; Do[tet = n (n + 1) (n + 2)/6; s = Floor[Sqrt[tet]]; If[s^2 + s == tet, AppendTo[t, tet]], {n, 0, 1000}]; t (* T. D. Noe, Mar 18 2013 *)
With[{nn=50000},Intersection[Binomial[Range[0,nn]+2,3],Table[n(n+1),{n,nn}]]] (* Harvey P. Dale, Apr 04 2016 *)

Python

def rootPronic(a):
    sr = 1<<33
    while a < sr*(sr+1):
      sr>>=1
    b = sr>>1
    while b:
        s = sr+b
        if a >= s*(s+1):
          sr = s
        b>>=1
    return sr
for i in range(1<<20):
      a = i*(i+1)*(i+2)//6
      t = rootPronic(a)
      if a == t*(t+1):
        print(a)

Extensions

fini, full keywords added by Max Alekseyev, Dec 28 2024