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Numbers that are both square and tetrahedral.
6

%I #31 Mar 01 2019 01:53:53

%S 0,1,4,19600

%N Numbers that are both square and tetrahedral.

%C A. J. J. Meyl proved in 1878 that only 1, 4 and 19600 are both square and tetrahedral. See link. [_Bernard Schott_, Dec 23 2012]

%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 600.

%D D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 165 (Rev. ed. 1997).

%H M. Gardner, <a href="/A001110/a001110.jpg">Letter to N. J. A. Sloane, circa Aug 11 1980</a>, concerning A001110, A027568, A039596, etc.

%H A. J. J. Meyl, <a href="http://archive.numdam.org/item/NAM_1878_2_17__464_1/">Question 1194</a>, Nouvelles Annales de Mathématiques, 2ème série, tome 17 (1878), p. 464-467.

%e From _Bernard Schott_, Dec 23 2012: (Start)

%e If S(n) = n^2 and T(m) = m*(m+1)*(m+2)/6, then

%e -> S(1)= T(1) = 1;

%e -> S(2)= T(2) = 4;

%e -> S(140) = T(48) = 19600. (End)

%t Select[Rest[FoldList[Plus, 0, Rest[FoldList[Plus, 0, Range[50000]]]]], IntegerQ[Sqrt[ # ]]&]

%t Intersection[Binomial[# + 2, 3]&/@Range[0, 10000], Range[0,409000]^2] (* _Harvey P. Dale_, Feb 01 2011 *)

%Y Intersection of A000290 and A000292.

%K nonn,fini,full

%O 1,3

%A _N. J. A. Sloane_