1,2

The problem arises when trying to build a square pyramid out of dominoes. The solution (m,k) = (3,7) for example corresponds to building a pyramid with layers of sizes 2 X 2, 4 X 4 and 6 X 6 from one set of double-6 dominoes.

The three nonzero solutions use one double-3 set, one double-6 set and one double-83 set. (The sequence 3,6,83 is too short to warrant a separate entry.)

The problem is equivalent to finding integers (m,k) such that 2m(m+1)(m+2)/3 = k*(k+1). This is a nonsingular cubic, so by Siegel's theorem, there are only finitely many solutions. - njas, May 25 2008. See also the articles by Stroeker and Tzanakis, and Stroeker and de Weger. (End)

J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer, 1992,

R. J. Stroeker and B. M. M. de Weger, Solving elliptic Diophantine equations: the general cubic case. Acta Arith. 87 (1999), 339-365.

R. J. Stroeker and N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. Acta Arith. 67 (1994), 177-196.

John Cannon, Using MAGMA to prove there are no other solutions

The known solutions are (m,k) = (0,0), (2,4), (3,7) and (17,84). There are no other solutions.

Simple-minded Maple program from njas:

f1:=m-> 1+8*m*(m+1)*(2*m+1)/3;

for m from 0 to 10^8 do if issqr(f1(m)) then lprint( m, (-1+sqrt(f1(m)))/2); fi; od;

Cf. A039596, A053611, A053612.

Adjacent sequences: A136273 A136274 A136275 this_sequence A136277 A136278 A136279

Sequence in context: A065674 A072954 A135790 this_sequence A024054 A126577 A073164

nonn,fini,full

Ken Knowlton (www.KnowltonMosaics.com), Mar 29 2008

Edited by njas, May 25 2008, Aug 17 2008

May 26 2008: John Cannon used MAGMA to show there are no further solutions (sse link)