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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(n-1) 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 + (x-y)^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^2-1 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(n-1) - 3*a(n-2) + 3*a(n-3). - Johannes W. Meijer, Oct 16 2009

G.f.: x*(2+5*x-x^2)/(1-x)^3. - Joerg Arndt, Sep 06 2014

a(n) = 6*n + a(n-1) - 3 for n > 1. - Vincenzo Librandi, Aug 08 2010

MAPLE

A080663 := proc(n) return 3*n^2-1: 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*x-1; print1(y" ") ) } checkit(n) = { for(x=1, n, y = 3*x*x-1; if(!issquare(y), print1(y" ")) ) }

(PARI) Vec(x*(2+5*x-x^2)/(1-x)^3+O(x^66)) \\ Joerg Arndt, Sep 06 2014

(MAGMA) [3*n^2-1 : 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

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Last modified December 13 09:58 EST 2019. Contains 329968 sequences. (Running on oeis4.)