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 A024494 a(n) = C(n,1) + C(n,4) + ... + C(n,3[n/3]+1). 15
 1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365, 2731, 5462, 10923, 21845, 43690, 87381, 174763, 349526, 699051, 1398101, 2796202, 5592405, 11184811, 22369622, 44739243, 89478485, 178956970, 357913941, 715827883, 1431655766, 2863311531, 5726623061, 11453246122 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 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 REFERENCES D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70. LINKS Index entries for linear recurrences with constant coefficients, signature (3,-3,2). FORMULA a(n) = (1/3)*(2^n+2*cos( (n-2)*Pi/3 )). G.f.: x(1-x)/((1-2x)(1-x+x^2)). - Paul Barry, Feb 11 2004 a(n) = sum{k=0..n, 2^k*2sin(-Pi*(n-k)/3+Pi/3)/sqrt(3)} (offset 0). - Paul Barry, May 18 2004 G.f.: (x*(1-x^2)*(1-x^3)/(1-x^6))/(1-2*x). - Michael Somos, Feb 14 2006 a(n+1)-2a(n) = A010892(n+1). - Michael Somos, Feb 14 2006 a(n) = 3a(n-1)-3a(n-2)+2a(n-3). - Paul Curtz, Nov 20 2007 Equals binomial transform of (1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1,...). - Gary W. Adamson, Jul 03 2008 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 MATHEMATICA 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 *) PROG (PARI) a(n) = sum(k=0, n\3, binomial(n, 3*k+1)) /* Michael Somos, Feb 14 2006 */ (PARI) a(n)=if(n<0, 0, ([1, 0, 1; 1, 1, 0; 0, 1, 1]^n)[2, 1]) /* Michael Somos, Feb 14 2006 */ CROSSREFS Cf. A010892. See A131708 for another version. Sequence in context: A300550 A014626 A132418 * A131708 A002991 A218532 Adjacent sequences:  A024491 A024492 A024493 * A024495 A024496 A024497 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified August 6 15:25 EDT 2020. Contains 336248 sequences. (Running on oeis4.)