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.

A165892 Triangular numbers of form n(n+2)(n+4).

Original entry on oeis.org

0, 15, 105, 2145, 32640, 73920, 1906128, 2299440, 7692030528
Offset: 1

Views

Author

Zak Seidov, Sep 29 2009

Keywords

Comments

Values of (m^2-1)/8 corresponding to ordinates of integral points on the elliptic curve: m^2 = 8*n^3 + 48*n^2 + 64*n + 1.
Corresponding values of n are given in A165893.

Crossrefs

Cf. A001219 Triangular numbers of form a(a+1)(a+2).

Programs

  • Mathematica
    TNQ[m_]:=IntegerQ[Sqrt[1+8*m]];Do[If[TNQ[m=n*(n+2)*(n+4)],Print[m]],{n,2*10^3}]
    Select[Table[n(n+2)(n+4),{n,0,2000}],OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Feb 07 2015 *)

Extensions

"fini", "full" keywords from Max Alekseyev, Oct 01 2009
Initial 0 added by Zak Seidov, Oct 04 2009 at the suggestion of Alexander R. Povolotsky.

A238008 Numbers n such that n*(n+3)*(n+6) is a triangular number.

Original entry on oeis.org

-5, -3, 0, 1, 10, 12, 22, 159, 639, 651, 2629
Offset: 1

Views

Author

Alex Ratushnyak, Feb 16 2014

Keywords

Comments

From Joerg Arndt, Feb 27 2014: (Start)
Using x for n, we are looking for integral points on the elliptic curve y*(y+1) == 2 * x*(x+3)*(x+6).
Substituting x --> x/2 and y --> y/2 and dividing the equation by 4 we obtain the Weierstrass form y^2 + 2*y == x^3 + 18*x^2 + 72*x.
Running the Sage program gives the following list of points (x : y : 1):
[(-10 : 8 : 1), (-7 : 5 : 1), (-6 : 0 : 1), (0 : 0 : 1), (2 : 14 : 1), (20 : 128 : 1), (24 : 160 : 1), (29 : 203 : 1), (44 : 350 : 1), (318 : 5830 : 1), (1278 : 46008 : 1), (1302 : 47304 : 1), (5258 : 381920 : 1)].
Dividing all x by 2 gives
[-5, -7/2, -3, 0, 1, 10, 12, 29/2, 22, 159, 639, 651, 2629].
The integral values are the terms of this sequence.
(End)

Examples

			1 is in the sequence because 1*4*7=28 is a triangular number.
		

Crossrefs

Programs

  • PARI
    istriang(n)=issquare(8*n+1);
    isok(n)=istriang( n*(n+3)*(n+6) );
    for (n=-10^6, 10^6, if ( isok(n), print1(n,", ") ) );
    \\ Joerg Arndt, Feb 17 2014
    
  • PARI
    isok(n) = ispolygonal((n-1)*n*(n+1), 3); \\ Michel Marcus, Mar 05 2014
  • Sage
    # for the curve y^2 + b1*x*y + b3*y =  x^3 + b2*x^2 + b4*x + b5, use
    # EllipticCurve([b1,b2, b3,b4,b5])
    # we have       y^2 +           2*y == x^3 + 18*x^2 + 72*x +  0, so need
    E=EllipticCurve([0, 18, 2, 72, 0])
    E.integral_points()
    ## Joerg Arndt, Feb 27 2014
    

Extensions

Added the negative terms, Joerg Arndt, Feb 27 2014
Showing 1-2 of 2 results.