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A027568
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Numbers that are both triangular and tetrahedral.
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18
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OFFSET
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1,3
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COMMENTS
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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)
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REFERENCES
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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.
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LINKS
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MAPLE
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{seq(binomial(i, 3), i=0..100000) } intersect {seq(binomial(k, 2), k= 0..100000)}; # Zerinvary Lajos, Apr 26 2008
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MATHEMATICA
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With[{trno=Accumulate[Range[0, 1000]]}, Intersection[trno, Accumulate[ trno]]] (* Harvey P. Dale, May 25 2014 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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