A102349 -id:A102349 - cp's OEIS frontend

A027568 Numbers that are both triangular and tetrahedral.

0, 1, 10, 120, 1540, 7140

Offset: 1

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)

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.

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

Intersection of A000217 and A000292.

Cf. A102349, A102461, A102772, A224421.

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

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

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

A224421 The indices of tetrahedral numbers that are also triangular.

Original entry on oeis.org

0, 1, 3, 8, 20, 34

Offset: 0

Tanya Khovanova, Apr 06 2013

			The eighth tetrahedral number is 120, which is also a triangular number. Hence, 8 belongs to the sequence.

E. T. Avanesov, Solution of a problem on figurate numbers (in Russian), Acta Arith. 12 1966/1967 pages 409-420.

Cf. A027568, A102349.

Select[Table[{n,(n(n+1)(n+2))/6},{n,0,35}],OddQ[Sqrt[1+8#[[2]]]]&][[;;,1]] (* Harvey P. Dale, May 31 2024 *)

for(n=0,34,if(ispolygonal(n*(n+1)*(n+2)/6,3),print1(n", "))) \\ Charles R Greathouse IV, Apr 07 2013

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A027568 Numbers that are both triangular and tetrahedral.

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A224421 The indices of tetrahedral numbers that are also triangular.

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