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%I #27 Jun 13 2015 00:54:42
%S -1,3,2,3,7,10,15,27,42,67,111,178,287,467,754,1219,1975,3194,5167,
%T 8363,13530,21891,35423,57314,92735,150051,242786,392835,635623,
%U 1028458,1664079,2692539,4356618,7049155,11405775,18454930,29860703,48315635,78176338,126491971
%N a(0)=-1, a(1)=3; a(n+2) = a(n+1) + a(n) + 2*A057078(n+1).
%C a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.
%C a(n) and its differences:
%C . -1, 3, 2, 3, 7, 10, 15, 27, 42,
%C . 4, -1, 1, 4, 3, 5, 12, 15, 25,
%C . -5, 2, 3, -1, 2, 7, 3, 10, 19,
%C . 7, 1, -4, 3, 5, -4, 7, 9, 4,
%C . -6, -5, 7, 2, -9, 11, 2, -5, 15,
%C . 1, 12, -5, -11, 20, -9, -7, 20, -5,
%C . 11, -17, -6, 31, -29, 2, 27, -25, 2,
%C . -28, 11, 37, -60, 31, 25, -52, 27, 29,
%C . 39, 26, -97, 91, -6, -77, 79, 2, -81.
%C Main diagonal: -(-1)^floor(n/2)*A108411(n).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2,1).
%F a(3n) = 2*F(3n)-1, a(3n+1) = 2*F(3n+1)+1, a(3n+2) = 2*F(3n+2), where F=A000045.
%F a(n+3) = a(n) + 4*F(n+1).
%F a(n) = A226328(n) + 1 for n>1.
%F a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) and many others by telescoping the fundamental recurrence.
%F G.f.: -(1-3*x-3*x^2-2*x^3) / ( (1-x-x^2)*(1+x+x^2) ). [_Bruno Berselli_, Jul 02 2013]
%F a(n) = a(n-2) + 2*a(n-3) - a(n-4). [_Bruno Berselli_, Jul 02 2013]
%e a(6) = 2*F(6)-1 = 2*8-1 = 15; a(7) = 2*F(7)+1 = 2*13+1 = 27; a(8) = 2*F(8) = 2*21 = 42.
%t a[n_] := (m = Mod[n, 3]; 2*Fibonacci[n] - (3*m - 1)*(m - 2)/2); Table[a[n], {n, 0, 39}] (* _Jean-François Alcover_, Jul 02 2013 *)
%Y Cf. A000045.
%K sign,easy
%O 0,2
%A _Paul Curtz_, Jul 01 2013
%E Edited by _Bruno Berselli_, Jul 02 2013
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