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%I #38 Sep 08 2022 08:45:27
%S 0,6,10,32,78,210,544,1430,3738,9792,25630,67106,175680,459942,
%T 1204138,3152480,8253294,21607410,56568928,148099382,387729210,
%U 1015088256,2657535550,6957518402,18215019648,47687540550,124847601994
%N Expansion of 2*x^2*(3-x)/((1+x)*(1-3*x+x^2)).
%C a(n) = the area of an irregular quadrilateral with vertices at the points (L(n),L(n+2)), (F(n+2),F(n+3)), (F(n+3),F(n+2)) and (L(n+2),L(n)), with F(n)=A000045(n) and L(n)=A000032(n). - _J. M. Bergot_, Jun 16 2014
%C a(n+1) appears also as the fourth component of the square of [F(n), F(n+1), F(n+2), F(n+3)], for n >= 0, where F(n) = A000045(n), in the Clifford algebra Cl_2 over Euclidean 2-space. The whole quartet of sequences for this square is [-A248161(n), A079472(n+1), A059929(n), a(n+1)]. See the Oct 15 2014 comment in A147973 where also a reference is given. - _Wolfdieter Lang_, Nov 01 2014
%H G. C. Greubel, <a href="/A121801/b121801.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).
%F a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
%F a(n) = -2*A121646(n+1).
%F G.f.: 2*x^2*(3-x)/((1+x)*(x^2-3*x+1)) (see name).
%F From _Wolfdieter Lang_, Nov 01 2014: (Start)
%F G.f.: (-10 + 8/(1+x) + 2*(1+x)/(1-3*x+x^2))/5 (partial fraction decomposition).
%F a(n) = (8*(-1)^n + 2*(F(2*(n+1)) + F(2*n)))/5 for n >= 1. a(0) = 0.
%F (End)
%F a(n) = 2*(Fibonacci(n)*Fibonacci(n+1) + (-1)^n). - _G. C. Greubel_, Jul 22 2019
%t c[i_, k_] := Floor[Mod[i/2^k, 2]] b[i_, k_] := If[c[i, k] == 0 && c[ i, k + 1] == 0, 0, If[c[i, k] == 1 && c[i, k + 1] == 1, 0, 1]] n = 4 - 1; M = Table[If[Sum[b[i, k]*b[j, k], {k, 0, n}] == 0, 1, 0], {j, 0, n}, {i, 0, n}] v[1] = {0, 1, 2, 3} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}] Det[M - x*IdentityMatrix[4]] Factor[%] aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[4]] == 0, x][[n]], {n, 1, 4}] Abs[aaa] a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}]
%t CoefficientList[Series[2*x*(3-x)/((1+x)*(1-3*x+x^2)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Jun 16 2014 *)
%t LinearRecurrence[{2,2,-1},{0,6,10},30] (* _Harvey P. Dale_, Jan 06 2015 *)
%t With[{F=Fibonacci}, Table[2*(F[n]*F[n+1] +(-1)^n), {n,30}]] (* _G. C. Greubel_, Jul 22 2019 *)
%o (PARI) concat(0,Vec(2*(3-x)/((1+x)*(1-3*x+x^2))+O(x^30))) \\ _Charles R Greathouse IV_, Sep 25 2012
%o (PARI) vector(30, n, f=fibonacci; 2*(f(n)*f(n+1)+(-1)^n) ) \\ _G. C. Greubel_, Jul 22 2019
%o (Magma) I:=[0,6,10]; [n le 3 select I[n] else 2*Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Nov 02 2014
%o (Magma) [2*(Lucas(2*n+1) +4*(-1)^n))/5: n in [1..30]]; // _G. C. Greubel_, Jul 22 2019
%o (Sage) [2*(lucas_number2(2*n+1,1,-1) +4*(-1)^n)/5 for n in (1..30)] # _G. C. Greubel_, Jul 22 2019
%o (GAP) List([1..30], n-> 2*(Lucas(1,-1,2*n+1)[2] +4*(-1)^n)/5 ); # _G. C. Greubel_, Jul 22 2019
%K nonn,easy
%O 1,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 27 2006
%E Edited by the Associate Editors of the OEIS, Aug 18 2009
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