%I
%S 2,11,26,47,74,107,146,191,242,299,362,431,506,587,674,767,866,971,
%T 1082,1199,1322,1451,1586,1727,1874,2027,2186,2351,2522,2699,2882,
%U 3071,3266,3467,3674,3887,4106,4331,4562,4799,5042,5291,5546,5807,6074,6347,6626
%N Numbers of the form 3*n^2  1.
%C These numbers cannot be perfect squares. See the link for a proof.
%C 2nd elementary symmetric polynomial of n, n + 1 and n + 2: n(n+1) + n(n+2) + (n+1)(n+2).  _Zak Seidov_, Mar 23 2005
%C a(n) = unsigned real term in (1 + ni)^3. E.g. (1 + 4i)^3 = (47  52i); where 52 = A121670(4). Note that (4 + i)^3 = (52 + 47i) = (A121670(4) + A080663(4)i).  _Gary W. Adamson_, Aug 14 2006
%C This sequence equals for n >= 2 the third right hand column of triangle A165674. Its recurrence relation leads to Pascal's triangle A007318. Crowley's formula for A080663(n1) leads to Wiggen's triangle A028421 and the o.g.f. of this sequence, without the first term, leads to Wood's polynomials A126671. See also A165676, A165677, A165678 and A165679.  _Johannes W. Meijer_, Oct 16 2009
%C The Diophantine equation x(x+1) + (x+2)(x+3) = (x+y)^2 + (xy)^2 has solutions x = a(n), y = 3n.  _Bruno Berselli_, Mar 29 2013
%C A simpler proof that these numbers can't be perfect squares can easily be constructed using congruences: If the equation x^2 = 3y^2  1 has a solution in positive integers, then x^2 = 2 mod 3. Obviously we can't have x = 0 mod 3, and x = 1 mod 3 doesn't work either because then x^2 = 1 mod 3 also. That leaves x = 2 mod 3, but then x^2 = 1 mod 3.  _Alonso del Arte_, Oct 19 2013
%C 2*a(n+1) = surface area of a rectangular prism with consecutive integer sides: n, n+1, and n+2, (n>0).  _Wesley Ivan Hurt_, Sep 06 2014
%D Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 7, Problem 6.6.
%D E. Grosswald, Topics from the Theory of Numbers, 1966 p 64 problem 11
%H Nathaniel Johnston, <a href="/A080663/b080663.txt">Table of n, a(n) for n = 1..10000</a>
%H Cino Hilliard, <a href="http://web.archive.org/web/20080411092534/http://groups.msn.com/BC2LCC/3n21isnotsquare.msnw">3n^21 not square</a>. [Archived copy as of Apr 11 2008 from web.archive.org]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SymmetricPolynomial.html">Symmetric Polynomial</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,1).
%F a(n) = 3*n^2  1.  _Stephen Crowley_, Jul 06 2009
%F a(n) = a(n1)  3*a(n2) + 3*a(n3).  _Johannes W. Meijer_, Oct 16 2009
%F G.f.: x*(2+5*xx^2)/(1x)^3.  _Joerg Arndt_, Sep 06 2014
%F a(n) = 6*n + a(n1)  3 for n > 1.  _Vincenzo Librandi_, Aug 08 2010
%p A080663 := proc(n) return 3*n^21: end proc: seq(A080663(n), n=1..50); # _Nathaniel Johnston_, Oct 16 2013
%t Table[Inner[Times, {n, n + 1, n + 2}, {n + 2, n + 3, n + 4}, Plus], {n, 47}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2010 *)
%t 3Range[47]^2  1 (* _Alonso del Arte_, Oct 19 2013 *)
%o (PARI) nosquare(n) = { for(x=1,n, y = 3*x*x1; print1(y" ") ) } checkit(n) = { for(x=1,n, y = 3*x*x1; if(!issquare(y),print1(y" ")) ) }
%o (PARI) Vec(x*(2+5*xx^2)/(1x)^3+O(x^66)) \\ _Joerg Arndt_, Sep 06 2014
%o (MAGMA) [3*n^21 : n in [1..50]]; // _Wesley Ivan Hurt_, Sep 04 2014
%Y Cf. A007318, A028421, A080663, A121670, A126671, A165674, A165676, A165677, A165678, A165679.
%K nonn,easy
%O 1,1
%A _Cino Hilliard_, Mar 01 2003
