A147973 - OEIS

%I #56 Sep 08 2022 08:45:38

%S -4,2,4,2,-4,-14,-28,-46,-68,-94,-124,-158,-196,-238,-284,-334,-388,

%T -446,-508,-574,-644,-718,-796,-878,-964,-1054,-1148,-1246,-1348,

%U -1454,-1564,-1678,-1796,-1918,-2044,-2174,-2308,-2446,-2588,-2734,-2884

%N a(n) = -2*n^2 + 12*n - 14.

%C -a(n+3) = 2*n^2 - 4, n >= 0, [-4,-2, 4, 14, ...] appears as the first member of the quartet for the square of [n, n+1, n+2, n+3], for n >= 0, in the Clifford algebra Cl_2. The other members are given in A046092(n), A054000(n+1) and A139570(n). The basis of Cl_2 is <1, s1, s2, s12> with s1.s1 = s2.s2 = 1, s12.s12 = -1, s1.s2 = -s2.s1 = s12. See e.g., pp. 5-6, eqs. (2.4)-(2.13) of the S. Gull et al. reference. - _Wolfdieter Lang_, Oct 15 2014

%C Related to the previous comment: if one uses the exterior (Grassmann) product with s1.s1 = s2.s2 = = s12.s12 = 0 and s1.s2 = -s2.s1 = s12, then the four components of the square of [n, n+1, n+2, n+3] are [A000290(n), A046092(n), A054000(n+1), A139570(n)], n >= 0. - _Wolfdieter Lang_, Nov 13 2014

%C 2 - a(n)/2 is a square. - _Bruno Berselli_, Apr 10 2018

%H Vincenzo Librandi, <a href="/A147973/b147973.txt">Table of n, a(n) for n = 1..1000</a>

%H S. Gull, A. Lasenby and C. Doran, <a href="http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are-not-real-the-geometric-algebra-of-spacetime/">Imaginary Numbers are not Real - the Geometric Algebra of Spacetime</a>, Found. Phys. 23(9), 1175-1201 (1993).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jul 10 2012

%F a(n) = -2*A008865(n-3). - _J. M. Bergot_, Jun 25 2018

%F G.f.: -2*x*(2 - 7*x + 7*x^2) / (1 - x)^3. - _Colin Barker_, Feb 12 2019

%p [-2*n^2+12*n-14$n=1..50]; # _Muniru A Asiru_, Feb 12 2019

%t lst={};Do[k=n^2-((n-1)^2+(n-2)^2+(n-3)^2);AppendTo[lst,k],{n,5!}];lst

%t Table[-2n^2+12n-14,{n,1,50}] (* _Vincenzo Librandi_, Jul 10 2012 *)

%t LinearRecurrence[{3,-3,1},{-4,2,4},50] (* _Harvey P. Dale_, Mar 02 2020 *)

%o (Magma) [-2*n^2+12*n-14: n in [1..50]]; // _Vincenzo Librandi_, Jul 10 2012

%o (PARI) a(n)=-2*n^2+12*n-14 \\ _Charles R Greathouse IV_, Sep 24 2015

%o (PARI) Vec(-2*x*(2 - 7*x + 7*x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, Feb 12 2019

%Y Cf. A008865.

%K sign,easy

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Nov 18 2008