

A080663


Numbers of the form 3*n^2  1.


15



2, 11, 26, 47, 74, 107, 146, 191, 242, 299, 362, 431, 506, 587, 674, 767, 866, 971, 1082, 1199, 1322, 1451, 1586, 1727, 1874, 2027, 2186, 2351, 2522, 2699, 2882, 3071, 3266, 3467, 3674, 3887, 4106, 4331, 4562, 4799, 5042, 5291, 5546, 5807, 6074, 6347, 6626
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

These numbers cannot be perfect squares. See the link for a proof.
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
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
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
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
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
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


REFERENCES

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 7, Problem 6.6.
E. Grosswald, Topics from the Theory of Numbers, 1966 p 64 problem 11


LINKS

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Cino Hilliard, 3n^21 not square. [Archived copy as of Apr 11 2008 from web.archive.org]
Eric Weisstein's World of Mathematics, Symmetric Polynomial
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = 3*n^2  1.  Stephen Crowley, Jul 06 2009
a(n) = a(n1)  3*a(n2) + 3*a(n3).  Johannes W. Meijer, Oct 16 2009
G.f.: x*(2+5*xx^2)/(1x)^3.  Joerg Arndt, Sep 06 2014
a(n) = 6*n + a(n1)  3 for n > 1.  Vincenzo Librandi, Aug 08 2010


MAPLE

A080663 := proc(n) return 3*n^21: end proc: seq(A080663(n), n=1..50); # Nathaniel Johnston, Oct 16 2013


MATHEMATICA

Table[Inner[Times, {n, n + 1, n + 2}, {n + 2, n + 3, n + 4}, Plus], {n, 47}] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
3Range[47]^2  1 (* Alonso del Arte, Oct 19 2013 *)


PROG

(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" ")) ) }
(PARI) Vec(x*(2+5*xx^2)/(1x)^3+O(x^66)) \\ Joerg Arndt, Sep 06 2014
(MAGMA) [3*n^21 : n in [1..50]]; // Wesley Ivan Hurt, Sep 04 2014


CROSSREFS

Cf. A007318, A028421, A080663, A121670, A126671, A165674, A165676, A165677, A165678, A165679.
Sequence in context: A077482 A141428 A104085 * A248118 A320648 A141464
Adjacent sequences: A080660 A080661 A080662 * A080664 A080665 A080666


KEYWORD

nonn,easy


AUTHOR

Cino Hilliard, Mar 01 2003


STATUS

approved



