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A071618 a(n+1) - 3*a(n) + a(n-1) = (2/3)(1+w^(n+1)+w^(2n+2)), where w = exp(2 Pi I / 3).

%I A071618 #24 Jun 13 2015 00:50:33
%S A071618 0,1,3,8,23,61,160,421,1103,2888,7563,19801,51840,135721,355323,
%T A071618 930248,2435423,6376021,16692640,43701901,114413063,299537288,
%U A071618 784198803,2053059121,5374978560,14071876561,36840651123,96450076808,252509579303
%N A071618 a(n+1) - 3*a(n) + a(n-1) = (2/3)(1+w^(n+1)+w^(2n+2)), where w = exp(2 Pi I / 3).
%C A071618 The sequence is closely related to the third term in the continued fraction expansion of 2(F(4n)+F(2n))/phi where F is the Fibonacci sequence. For any k smaller than a(n), k*F(2n)*phi has to be rounded by excess, for any k greater than a(n), k*F(2n)*phi has to be rounded by default. - _Thomas Baruchel_, Aug 31 2004
%H A071618 Harvey P. Dale, <a href="/A071618/b071618.txt">Table of n, a(n) for n = 0..1000</a>
%H A071618 S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, <a href="http://arXiv.org/abs/nlin.SI/0104020">Blending two discrete integrability criteria: ...</a>
%H A071618 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,1,-3,1).
%F A071618 a(n) = floor ( phi^2n / 2 ) = floor ( (Lucas(2n)-1) / 2 ). - _Thomas Baruchel_, Aug 31 2004
%F A071618 a(-n) = a(n). a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4) + 2. - _Michael Somos_, Mar 08 2007
%F A071618 G.f.: x*(1+x^3) / ((1-x^3)* (1-3*x+x^2)). - _Michael Somos_, Mar 08 2007
%F A071618 a(0)=0, a(1)=1, a(2)=3, a(3)=8, a(4)=23, a(n) = 3*a(n-1) - a(n-2) + a(n-3) - 3*a(n-4) + a(n-5). - _Harvey P. Dale_, Dec 18 2011
%t A071618 a[ -1 ] = 0; a[ 0 ] = 1; w = Exp[ 2Pi*I/3 ]; a[ n_ ] := a[ n ] = Simplify[ (2/3)(1 + w^n + w^(2n)) + 3a[ n - 1 ] - a[ n - 2 ] ]; Table[ a[ n ], {n, -1, 28} ]
%t A071618 LinearRecurrence[{3,-1,1,-3,1},{0,1,3,8,23},30] (* or *) CoefficientList[ Series[x (1+x^3)/((1-x^3)*(1-3x+x^2)),{x,0,30}],x] (* _Harvey P. Dale_, Dec 18 2011 *)
%o A071618 (PARI) u=0;v=1;for(n=1,30,print1(a=3*v-u+2*!(n%3)," ");u=v;v=a) /* _Thomas Baruchel_ */
%o A071618 (PARI) {a(n)= ( fibonacci(2*n+1)+ fibonacci(2*n-1)+ (n%3>0))/2- 1 } /* _Michael Somos_, Mar 08 2007 */
%o A071618 (PARI) {a(n)= n=abs(n); polcoeff( x*(1+x^3)/ ((1-x^3)* (1-3*x+x^2)) +x*O(x^n), n)} /* _Michael Somos_, Mar 08 2007 */
%K A071618 nonn,easy
%O A071618 0,3
%A A071618 _N. J. A. Sloane_, Jun 21 2002