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%I A027568 #78 Jul 22 2023 08:13:56
%S A027568 0,1,10,120,1540,7140
%N A027568 Numbers that are both triangular and tetrahedral.
%C A027568 From _Anthony C Robin_, Oct 27 2022: (Start)
%C A027568 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).
%C A027568 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)
%D A027568 J.-M. De Koninck, Ces nombres qui nous fascinent, Ellipses (Paris), 2008 (entry 10, page 3; entry 120, page 41).
%D A027568 L. J. Mordell, Diophantine Equations, Ac. Press, page 258.
%D A027568 P. Odifreddi, Il museo dei numeri, Rizzoli, 2014, page 224.
%D A027568 J. Roberts, The Lure of the Integers, page 53.
%H A027568 E. T. Avanesov, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa12/aa12128.pdf">Solution of a problem on figurate numbers</a> (in Russian), Acta Arith. 12 1966/1967 pages 409-420.
%H A027568 Patrick De Geest, <a href="http://www.worldofnumbers.com/tetra.htm">Palindromic Tetrahedrals</a>
%H A027568 M. Gardner, <a href="/A001110/a001110.jpg">Letter to N. J. A. Sloane, circa Aug 11 1980</a>, concerning A001110, A027568, A039596, etc.
%H A027568 J. Roberts, <a href="/A027568/a027568.pdf">The Lure of the Integers</a>, pp. 53. (Annotated scanned copy)
%H A027568 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>
%p A027568 {seq(binomial(i,3),i=0..100000) } intersect {seq(binomial(k,2), k= 0..100000)}; # _Zerinvary Lajos_, Apr 26 2008
%t A027568 With[{trno=Accumulate[Range[0,1000]]},Intersection[trno,Accumulate[ trno]]] (* _Harvey P. Dale_, May 25 2014 *)
%o A027568 (PARI) for(n=0,1e3,if(ispolygonal(t=n*(n+1)*(n+2)/6,3),print1(t", "))) \\ _Charles R Greathouse IV_, Apr 07 2013
%Y A027568 Intersection of A000217 and A000292.
%Y A027568 Cf. A102349, A102461, A102772, A224421.
%K A027568 nonn,fini,full
%O A027568 1,3
%A A027568 _Patrick De Geest_

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