OFFSET
1,2
COMMENTS
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
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
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
FORMULA
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
a(n) = -2*A121646(n+1).
G.f.: 2*x^2*(3-x)/((1+x)*(x^2-3*x+1)) (see name).
From Wolfdieter Lang, Nov 01 2014: (Start)
G.f.: (-10 + 8/(1+x) + 2*(1+x)/(1-3*x+x^2))/5 (partial fraction decomposition).
a(n) = (8*(-1)^n + 2*(F(2*(n+1)) + F(2*n)))/5 for n >= 1. a(0) = 0.
(End)
a(n) = 2*(Fibonacci(n)*Fibonacci(n+1) + (-1)^n). - G. C. Greubel, Jul 22 2019
MATHEMATICA
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}]
CoefficientList[Series[2*x*(3-x)/((1+x)*(1-3*x+x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 16 2014 *)
LinearRecurrence[{2, 2, -1}, {0, 6, 10}, 30] (* Harvey P. Dale, Jan 06 2015 *)
With[{F=Fibonacci}, Table[2*(F[n]*F[n+1] +(-1)^n), {n, 30}]] (* G. C. Greubel, Jul 22 2019 *)
PROG
(PARI) concat(0, Vec(2*(3-x)/((1+x)*(1-3*x+x^2))+O(x^30))) \\ Charles R Greathouse IV, Sep 25 2012
(PARI) vector(30, n, f=fibonacci; 2*(f(n)*f(n+1)+(-1)^n) ) \\ G. C. Greubel, Jul 22 2019
(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
(Magma) [2*(Lucas(2*n+1) +4*(-1)^n)/5: n in [1..30]]; // G. C. Greubel, Jul 22 2019
(Sage) [2*(lucas_number2(2*n+1, 1, -1) +4*(-1)^n)/5 for n in (1..30)] # G. C. Greubel, Jul 22 2019
(GAP) List([1..30], n-> 2*(Lucas(1, -1, 2*n+1)[2] +4*(-1)^n)/5 ); # G. C. Greubel, Jul 22 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula and Gary W. Adamson, Aug 27 2006
EXTENSIONS
Edited by the Associate Editors of the OEIS, Aug 18 2009
STATUS
approved