OFFSET
1,3
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.
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
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 *)
PROG
(PARI) for(n=0, 1e3, if(ispolygonal(t=n*(n+1)*(n+2)/6, 3), print1(t", "))) \\ Charles R Greathouse IV, Apr 07 2013
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
STATUS
approved