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A136276 Consider pairs of nonnegative integers (m,k) such that 2^2 + 4^2 + 6^2 + ... + (2m)^2 = k(k+1); sequence gives k values.

%I A136276 #12 Aug 17 2025 03:59:07
%S A136276 0,4,7,84
%N A136276 Consider pairs of nonnegative integers (m,k) such that 2^2 + 4^2 + 6^2 + ... + (2m)^2 = k(k+1); sequence gives k values.
%C A136276 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.
%C A136276 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.)
%C A136276 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. - _N. J. A. Sloane_, May 25 2008. See also the articles by Stroeker and Tzanakis and Stroeker and de Weger.
%H A136276 John Cannon, <a href="/A136276/a136276.txt">Using MAGMA to prove there are no other solutions</a>
%H A136276 J. H. Silverman and J. Tate, <a href="http://dx.doi.org/10.1007/978-1-4757-4252-7">Rational Points on Elliptic Curves</a>, Springer, 1992,
%H A136276 R. J. Stroeker and B. M. M. de Weger, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa87/aa8743.pdf">Solving elliptic Diophantine equations: the general cubic case</a>, Acta Arith. 87 (1999), 339-365.
%H A136276 R. J. Stroeker and N. Tzanakis, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa67/aa6725.pdf">Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms</a>, Acta Arith. 67 (1994), 177-196.
%e A136276 The known solutions are (m,k) = (0,0), (2,4), (3,7) and (17,84). There are no other solutions.
%p A136276 Simple-minded Maple program from _N. J. A. Sloane_:
%p A136276 f1:=m-> 1+8*m*(m+1)*(2*m+1)/3;
%p A136276 for m from 0 to 10^8 do if issqr(f1(m)) then lprint( m, (-1+sqrt(f1(m)))/2); fi; od;
%Y A136276 Cf. A039596, A053611, A053612.
%K A136276 nonn,fini,full
%O A136276 1,2
%A A136276 Ken Knowlton (www.KnowltonMosaics.com), Mar 29 2008
%E A136276 Edited by _N. J. A. Sloane_, May 25 2008, Aug 17 2008
%E A136276 May 26 2008: John Cannon used MAGMA to show there are no further solutions (see link)