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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. 1
0, 4, 7, 84 (list; graph; refs; listen; history; text; internal format)



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. - N. J. A. Sloane, May 25 2008. See also the articles by Stroeker and Tzanakis and Stroeker and de Weger. (End)


Table of n, a(n) for n=1..4.

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

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.


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 N. J. A. Sloane:

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.

Sequence in context: A135790 A253202 A156474 * A072954 A322723 A220003

Adjacent sequences:  A136273 A136274 A136275 * A136277 A136278 A136279




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


Edited by N. J. A. Sloane, May 25 2008, Aug 17 2008

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



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