cp's OEIS Frontend

A227104 a(0)=-1, a(1)=3; a(n+2) = a(n+1) + a(n) + 2*A057078(n+1).

%I A227104 #27 Jun 13 2015 00:54:42
%S A227104 -1,3,2,3,7,10,15,27,42,67,111,178,287,467,754,1219,1975,3194,5167,
%T A227104 8363,13530,21891,35423,57314,92735,150051,242786,392835,635623,
%U A227104 1028458,1664079,2692539,4356618,7049155,11405775,18454930,29860703,48315635,78176338,126491971
%N A227104 a(0)=-1, a(1)=3; a(n+2) = a(n+1) + a(n) + 2*A057078(n+1).
%C A227104 a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.
%C A227104 a(n) and its differences:
%C A227104 .  -1,   3,   2,   3,   7,  10,  15,  27,  42,
%C A227104 .   4,  -1,   1,   4,   3,   5,  12,  15,  25,
%C A227104 .  -5,   2,   3,  -1,   2,   7,   3,  10,  19,
%C A227104 .   7,   1,  -4,   3,   5,  -4,   7,   9,   4,
%C A227104 .  -6,  -5,   7,   2,  -9,  11,   2,  -5,  15,
%C A227104 .   1,  12,  -5, -11,  20,  -9,  -7,  20,  -5,
%C A227104 .  11, -17,  -6,  31, -29,   2,  27, -25,   2,
%C A227104 . -28,  11,  37, -60,  31,  25, -52,  27,  29,
%C A227104 .  39,  26, -97,  91,  -6, -77,  79,   2, -81.
%C A227104 Main diagonal: -(-1)^floor(n/2)*A108411(n).
%H A227104 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2,1).
%F A227104 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 A227104 a(n+3) = a(n) + 4*F(n+1).
%F A227104 a(n) = A226328(n) + 1 for n>1.
%F A227104 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 A227104 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 A227104 a(n) = a(n-2) + 2*a(n-3) - a(n-4). [_Bruno Berselli_, Jul 02 2013]
%e A227104 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 A227104 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 A227104 Cf. A000045.
%K A227104 sign,easy
%O A227104 0,2
%A A227104 _Paul Curtz_, Jul 01 2013
%E A227104 Edited by _Bruno Berselli_, Jul 02 2013