A080663 - OEIS

%I #76 Aug 23 2024 08:40:09

%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 a(n) = 3*n^2 - 1.

%C These numbers cannot be perfect squares. See the Hilliard 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 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

%C 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

%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) is surface area of a rectangular prism with consecutive integer sides: n, n+1, and n+2, (n>0). - _Wesley Ivan Hurt_, Sep 06 2014

%C Numbers m such that 3*m+3 is a square. So, these are the numbers m such that the system of equations x=sqrt(m-2yz), y=sqrt(m+1-2xz), z=sqrt(m+2-2xy) admits 3 real positive solutions whose sum is an integer. See the Rechtman link. - _Michel Marcus_, Jun 06 2020

%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^2 - 1 not square</a>. [Archived copy as of Apr 11 2008 from web.archive.org]

%H Ana Rechtman, <a href="http://images.math.cnrs.fr/Juin-2020-1er-defi.html">Juin 2020, 1er défi</a>, Images des Mathématiques, CNRS, 2020 (in French).

%H Leo Tavares, <a href="/A080663/a080663.jpg">Illustration: Conjoined Trapezoids</a>

%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) = -Re((1 + n*i)^3) where i=sqrt(-1). - _Gary W. Adamson_, Aug 14 2006

%F a(n) = 3*n^2 - 1. - _Stephen Crowley_, Jul 06 2009

%F a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3). - _Johannes W. Meijer_, Oct 16 2009

%F G.f.: x*(2 + 5*x - x^2)/(1-x)^3. - _Joerg Arndt_, Sep 06 2014

%F a(n) = a(n-1) + 6*n - 3 for n > 1. - _Vincenzo Librandi_, Aug 08 2010

%F E.g.f.: 1 + exp(x)*(3*x^2 + 3*x - 1). - _Stefano Spezia_, Feb 01 2020

%F From _Amiram Eldar_, Feb 04 2021: (Start)

%F Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(3))*cot(Pi/sqrt(3)))/2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(3))*csc(Pi/sqrt(3)) - 1)/2.

%F Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(3))*csc(Pi/sqrt(3)).

%F Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(3))*sin(sqrt(2/3)*Pi)/sqrt(2). (End)

%F a(n) = A005449(n) + A115067(n). - _Leo Tavares_, May 25 2022

%F a(n) = (n-1)*n + (n-1)*(n+1) + n*(n+1), for n >= 1. See the _Zak Seidov_ comment above. - _Wolfdieter Lang_, Aug 15 2024

%p A080663 := proc(n) return 3*n^2-1: end proc: seq(A080663(n), n=1..50); # _Nathaniel Johnston_, Oct 16 2013

%t 3*Range[47]^2 - 1 (* _Alonso del Arte_, Oct 19 2013 *)

%o (PARI) list(n) = { for(x=1,n, y = 3*x*x-1; print1(y, ", ") ) } \\ edited by _Michel Marcus_, Feb 01 2020

%o (PARI) Vec(x*(2+5*x-x^2)/(1-x)^3+O(x^66)) \\ _Joerg Arndt_, Sep 06 2014

%o (Magma) [3*n^2-1 : n in [1..50]]; // _Wesley Ivan Hurt_, Sep 04 2014

%Y Cf. A007318, A028421, A080663, A121670, A126671, A165674, A165676, A165677, A165678, A165679.

%Y Cf. A005449, A115067.

%K nonn,easy

%O 1,1

%A _Cino Hilliard_, Mar 01 2003