OFFSET
1,2
COMMENTS
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
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
G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).
FORMULA
3*a(n) = 2^n + 2*cos( (n-2)*Pi/3 )) = 2^n - A057079(n+2).
G.f.: x*(1-x)/((1-2*x)*(1-x+x^2)). - Paul Barry, Feb 11 2004
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
G.f.: (x*(1-x^2)*(1-x^3)/(1-x^6))/(1-2*x). - Michael Somos, Feb 14 2006
a(n+1) - 2*a(n) = A010892(n+1). - Michael Somos, Feb 14 2006
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(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 *)
LinearRecurrence[{3, -3, 2}, {1, 2, 3}, 40] (* G. C. Greubel, Jan 23 2023 *)
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 */
(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
(SageMath)
def A024494(n): return (1/3)*(2^n -chebyshev_U(n, 1/2) +2*chebyshev_U(n-1, 1/2))
[A024494(n) for n in range(1, 41)] # G. C. Greubel, Jan 23 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved