A147973 a(n) = -2*n^2 + 12*n - 14.

-4, 2, 4, 2, -4, -14, -28, -46, -68, -94, -124, -158, -196, -238, -284, -334, -388, -446, -508, -574, -644, -718, -796, -878, -964, -1054, -1148, -1246, -1348, -1454, -1564, -1678, -1796, -1918, -2044, -2174, -2308, -2446, -2588, -2734, -2884, -3038, -3196, -3358

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Nov 18 2008

Keywords

• sign
• easy

Comments

-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

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

2 - a(n)/2 is a square. - Bruno Berselli, Apr 10 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• S. Gull, A. Lasenby and C. Doran, Imaginary Numbers are not Real - the Geometric Algebra of Spacetime, Found. Phys., Vol. 23(9) (1993), pp. 1175-1201.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A008865, A046092, A054000, A139570.

Programs

Magma

[-2*n^2+12*n-14: n in [1..50]]; // Vincenzo Librandi, Jul 10 2012

Maple

[-2*n^2+12*n-14$n=1..50]; # Muniru A Asiru, Feb 12 2019

Mathematica

lst={};Do[k=n^2-((n-1)^2+(n-2)^2+(n-3)^2);AppendTo[lst,k],{n,5!}];lst
Table[-2n^2+12n-14,{n,1,50}] (* Vincenzo Librandi, Jul 10 2012 *)
LinearRecurrence[{3,-3,1},{-4,2,4},50] (* Harvey P. Dale, Mar 02 2020 *)

PARI

a(n)=-2*n^2+12*n-14 \\ Charles R Greathouse IV, Sep 24 2015

PARI

Vec(-2*x*(2 - 7*x + 7*x^2) / (1 - x)^3 + O(x^40)) \\ Colin Barker, Feb 12 2019

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 10 2012

a(n) = -2*A008865(n-3). - J. M. Bergot, Jun 25 2018

G.f.: -2*x*(2 - 7*x + 7*x^2)/(1 - x)^3. - Colin Barker, Feb 12 2019

E.g.f.: -2*(exp(x)*(x^2 - 5*x + 7) - 7). - Elmo R. Oliveira, Nov 17 2024