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A024494 a(n) = C(n,1) + C(n,4) + ... + C(n,3[n/3]+1). 14

%I

%S 1,2,3,5,10,21,43,86,171,341,682,1365,2731,5462,10923,21845,43690,

%T 87381,174763,349526,699051,1398101,2796202,5592405,11184811,22369622,

%U 44739243,89478485,178956970,357913941,715827883,1431655766,2863311531,5726623061,11453246122

%N a(n) = C(n,1) + C(n,4) + ... + C(n,3[n/3]+1).

%C M^n * [1,0,0] = [A024493(n), A024495(n), a(n)], where M = a 3x3 matrix [1,1,0; 0,1,1; 1,0,1]. Sum of terms = 2^n. Example: M^5 * [1,0,0] = [11, 11, 10], sum = 2^5 = 32. - _Gary W. Adamson_, Mar 13 2009

%C Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1+M)^n = A024493(n)+a(n)*M+A024495(n)*M^2. - _Stanislav Sykora_, Jun 10 2012

%D D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2).

%F a(n) = (1/3)*(2^n+2*cos( (n-2)*Pi/3 )).

%F G.f.: x(1-x)/((1-2x)(1-x+x^2)). - _Paul Barry_, Feb 11 2004

%F a(n) = sum{k=0..n, 2^k*2sin(-Pi*(n-k)/3+Pi/3)/sqrt(3)} (offset 0). - _Paul Barry_, May 18 2004

%F G.f.: (x*(1-x^2)*(1-x^3)/(1-x^6))/(1-2*x). - _Michael Somos_, Feb 14 2006

%F a(n+1)-2a(n) = A010892(n+1). - _Michael Somos_, Feb 14 2006

%F a(n) = 3a(n-1)-3a(n-2)+2a(n-3). - _Paul Curtz_, Nov 20 2007

%F Equals binomial transform of (1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1,...). - _Gary W. Adamson_, Jul 03 2008

%F Start with x(0)=1,y(0)=0,z(0)=0 and set x(n+1)=x(n)+z(n), y(n+1)=y(n)+x(n),z(n+1)=z(n)+y(n). Then a(n)=y(n). - _Stanislav Sykora_, Jun 10 2012

%t nn=20;a=1/(1-x);Drop[CoefficientList[Series[a x /(1-x-x^3 a^2),{x,0,nn}],x],1] (* _Geoffrey Critzer_, Dec 22 2013 *)

%o (PARI) a(n) = sum(k=0,n\3,binomial(n,3*k+1)) /* _Michael Somos_, Feb 14 2006 */

%o (PARI) a(n)=if(n<0, 0, ([1,0,1;1,1,0;0,1,1]^n)[2,1]) /* _Michael Somos_, Feb 14 2006 */

%Y Cf. A010892. See A131708 for another version.

%K nonn,easy,changed

%O 1,2

%A _Clark Kimberling_

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Last modified March 20 05:17 EDT 2019. Contains 321344 sequences. (Running on oeis4.)