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%I #52 Jan 28 2023 12:33:10
%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*floor(n/3) + 1).
%C M^n * [1,0,0] = [A024493(n), A024495(n), a(n)], where M is a 3 X 3 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 G. C. Greubel, <a href="/A024494/b024494.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2).
%F 3*a(n) = 2^n + 2*cos( (n-2)*Pi/3 )) = 2^n - A057079(n+2).
%F G.f.: x*(1-x)/((1-2*x)*(1-x+x^2)). - _Paul Barry_, Feb 11 2004
%F a(n) = Sum_{k=0..n} 2^k*2*sin(-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) - 2*a(n) = A010892(n+1). - _Michael Somos_, Feb 14 2006
%F a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(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 *)
%t LinearRecurrence[{3,-3,2}, {1,2,3}, 40] (* _G. C. Greubel_, Jan 23 2023 *)
%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 */
%o (Magma) [n le 3 select n else 3*Self(n-1) -3*Self(n-2) +2*Self(n-3): n in [1..40]]; // _G. C. Greubel_, Jan 23 2023
%o (SageMath)
%o def A024494(n): return (1/3)*(2^n -chebyshev_U(n,1/2) +2*chebyshev_U(n-1,1/2))
%o [A024494(n) for n in range(1,41)] # _G. C. Greubel_, Jan 23 2023
%Y Cf. A010892, A024493, A024495, A057079.
%Y See A131708 for another version.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_
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