A027568 Numbers that are both triangular and tetrahedral.

0, 1, 10, 120, 1540, 7140

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• fini
• full

Comments

From Anthony C Robin, Oct 27 2022: (Start)

For numbers to be triangular and tetrahedral, we look for solutions r*(r+1)*(r+2)/6 = t*(t+1)/2 = a(n). The corresponding r and t are r = A224421(n-1) and t = A102349(n).

Writing m=r+1 and s=2t+1, this problem is equivalent to solving the Diophantine equation 3 + 4*(m^3 - m) = 3*s^2. The integer solutions for this equation are m = 0, 1, 2, 4, 9, 21, 35 and the corresponding values of s are 1, 1, 3, 9, 31, 111, 239. (End)

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Ellipses (Paris), 2008 (entry 10, page 3; entry 120, page 41).
• L. J. Mordell, Diophantine Equations, Ac. Press, page 258.
• P. Odifreddi, Il museo dei numeri, Rizzoli, 2014, page 224.
• J. Roberts, The Lure of the Integers, page 53.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 21.

Links

• E. T. Avanesov, Solution of a problem on figurate numbers (in Russian), Acta Arith. 12 1966/1967 pages 409-420.
• Patrick De Geest, Palindromic Tetrahedrals
• M. Gardner, Letter to N. J. A. Sloane, circa Aug 11 1980, concerning A001110, A027568, A039596, etc.
• J. Roberts, The Lure of the Integers, pp. 53. (Annotated scanned copy)
• Eric Weisstein's World of Mathematics, Tetrahedral Number

Crossrefs

Intersection of A000217 and A000292.

Cf. A102349, A102461, A102772, A224421.

Programs

Maple

{seq(binomial(i,3),i=0..100000) } intersect {seq(binomial(k,2), k= 0..100000)}; # Zerinvary Lajos, Apr 26 2008

Mathematica

With[{trno=Accumulate[Range[0,1000]]},Intersection[trno,Accumulate[ trno]]] (* Harvey P. Dale, May 25 2014 *)

PARI

for(n=0,1e3,if(ispolygonal(t=n*(n+1)*(n+2)/6,3),print1(t", "))) \\ Charles R Greathouse IV, Apr 07 2013