

A174592


Numbers n such that n^2 + 2*(n+2)^2 is a square.


1



2, 46, 658, 9182, 127906, 1781518, 24813362, 345605566, 4813664578, 67045698542, 933826115026, 13006519911838, 181157452650722, 2523197817198286, 35143611988125298, 489487370016555902, 6817679568243657346
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

The equation n^2 + 2*(n+2)^2 = y^2 is transformed via x=3n+4 into the Diophantine equation x^2  3*y^2 = 8, and by division through 4 to (x/2)^2  3*(y/2)^2 = 2. Setting xbar = x/2 and ybar = y/2, the fundamental solution to xbar^2  3*ybar^2 = 2 is xbar = ybar = 1, and the general solution is given by multiplying (1+sqrt(3))*(u+sqrt(3)*v)^j, j=1,2,3,4,... where (u,v) = (A001075(j), A001353(j)). Expanding this product, isolating the square root., etc., and discarding the solutions that are associated with noninteger n generates the series of all solutions.  R. J. Mathar, May 02 2010
Also numbers n such that the sum of the four pentagonal numbers starting at index n is equal to the sum of four consecutive triangular numbers.  Colin Barker, Dec 19 2014


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..890
Index entries for linear recurrences with constant coefficients, signature (15,15,1).


FORMULA

From Bruno Berselli, Sep 07 2011: (Start)
G.f.: 2*x*(1+8*xx^2)/((1x)*(114*x+x^2)).
a(n) = 15*a(n1)  15*a(n2) + a(n3).
a(n) = 4/3 + ((sqrt(3)+1)^(4n1)  (sqrt(3)1)^(4n1))/(3*2^(2n1)). (End)


MATHEMATICA

eq = Simplify[n^2 + 2*(n+2)^2 == y^2 /. n > (x  4)/3]; r = Reduce[x >= 0 && y >= 0 && eq, x, Integers] /. C[1] > k; xx[k_] = x /. ToRules[r[[1, 1]]]; Select[Table[Simplify[(xx[k]  4)/3], {k, 1, 34}], IntegerQ] (* JeanFrançois Alcover, Sep 06 2011, after R. J. Mathar *)
CoefficientList[Series[2 (1 + 8 x  x^2)/((1  x) (1 14 x + x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 24 2014 °)


PROG

(MAGMA) [n: n in [0..70000000]  IsSquare(3*n^2+8*n+8)];
(MAGMA) I:=[2, 46, 658]; [n le 3 select I[n] else 15*Self(n1)15*Self(n2)+Self(n3): n in [1..17]]; // Bruno Berselli, Sep 07 2011


CROSSREFS

Sequence in context: A302377 A303098 A302949 * A066555 A290046 A012001
Adjacent sequences: A174589 A174590 A174591 * A174593 A174594 A174595


KEYWORD

nonn,easy


AUTHOR

Vincenzo Librandi, Apr 11 2010


EXTENSIONS

More terms from R. J. Mathar, May 02 2010


STATUS

approved



