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.

A216268 Tetrahedral numbers of the form k^2 - 1.

Original entry on oeis.org

0, 35, 120, 2024, 2600, 43680, 435730689800
Offset: 1

Views

Author

Alex Ratushnyak, Mar 15 2013

Keywords

Comments

This sequence is finite by Siegel's theorem on integral points. The next term, if it exists, is greater than 10^35. - David Radcliffe, Jan 01 2024

Crossrefs

Cf. A003556 (both square and tetrahedral).

Programs

  • Maple
    select(t -> issqr(t+1), [seq(i*(i+1)*(i+2)/6, i=0..10^6)]); # Robert Israel, Jan 02 2024
  • Mathematica
    t = {}; Do[tet = n (n + 1) (n + 2)/6; If[IntegerQ[Sqrt[tet + 1]], AppendTo[t, tet]], {n, 0, 100000}]; t (* T. D. Noe, Mar 18 2013 *)
  • PARI
    A000292(n) = n*(n+1)*(n+2)\6;
    for(n=0,10^9, t=A000292(n); if (issquare(t+1), print1(t,", ") ) );
    /* Joerg Arndt, Mar 16 2013 */
  • Python
    import math
    for i in range(1<<33):
        t = i*(i+1)*(i+2)//6 + 1
        sr = math.isqrt(t)
        if sr*sr == t:
            print (t-1, sep=' ')
    

A175492 Numbers m >= 3 such that binomial(m,3) + 1 is a square.

Original entry on oeis.org

7, 10, 24, 26, 65, 13777
Offset: 1

Views

Author

Ctibor O. Zizka, May 29 2010

Keywords

Comments

Related sequences:
Numbers m such that binomial(m,2) is a square: A055997;
Numbers m such that binomial(m,2) + 1 is a square: A006451 + 1;
Numbers m such that binomial(m,2) - 1 is a square: A072221 + 1;
Numbers m >= 3 such that binomial(m,3) is a square: {3, 4, 50} (Proved by A. J. Meyl in 1878);
Numbers m >= 4 such that binomial(m,4) + 1 is a square: {6, 7, 45, 55, ...};
Numbers m >= 7 such that binomial(m,7) + 1 is a square: {8, 10, 21, 143, ...}.
No additional terms up to 10 million. - Harvey P. Dale, Apr 04 2017
No additional terms up to 10 billion. - Jon E. Schoenfield, Mar 18 2022
No additional terms up to 1 trillion. The sequence is finite by Siegel's theorem on integral points. - David Radcliffe, Jan 01 2024

Crossrefs

Cf. A216268 (values of binomial(m, 3)) and A216269 (square roots of binomial(m, 3) + 1).

Programs

  • Mathematica
    lst = {}; k = 3; While[k < 10^6, If[ IntegerQ@ Sqrt[ Binomial[k, 3] + 1], AppendTo[lst, k]]; k++ ]; lst (* Robert G. Wilson v, Jun 11 2010 *)
    Select[Range[3,14000],IntegerQ[Sqrt[Binomial[#,3]+1]]&] (* Harvey P. Dale, Apr 04 2017 *)
  • PARI
    isok(m) = (m>=3) && issquare(binomial(m,3)+1); \\ Michel Marcus, Mar 15 2022
    
  • Python
    from sympy import binomial
    from sympy.ntheory.primetest import is_square
    for m in range(3, 10**6):
        if is_square(binomial(m,3)+1):
            print(m) # Mohammed Yaseen, Mar 18 2022
Showing 1-2 of 2 results.