

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




OFFSET

1,2


COMMENTS

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 double6 dominoes.
The three nonzero solutions use one double3 set, one double6 set and one double83 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)


LINKS

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), 339365.
R. J. Stroeker and N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994), 177196.


EXAMPLE

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


MAPLE

Simpleminded 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;


CROSSREFS

Cf. A039596, A053611, A053612.
Sequence in context: A135790 A253202 A156474 * A072954 A322723 A220003
Adjacent sequences: A136273 A136274 A136275 * A136277 A136278 A136279


KEYWORD

nonn,fini,full


AUTHOR

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


EXTENSIONS

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)


STATUS

approved



