0,4

First differences of A131708. First differences give A024495. - Paul Curtz (bpcrtz(AT)free.fr), Nov 18 2007

a(n) = upper left term of X^n, where X = the 4 X 4 matrix [1,0,1,0; 1,1,0,0; 0,1,1,1; 0,0,0,1]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 01 2008

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 13 2009: (Start)

M^n * [1,0,0] = [a(n), A024495(n), A024494(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. (End)

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

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

G.f.: (1-x)^2/((1-2x)(1-x+x^2))=(1-2x+x^2)/(1-3x+3x^2-2x^3) - Paul Barry (pbarry(AT)wit.ie), Feb 11 2004

a(n)=(1/3)*(2^n+b(n)) where b(n) is the 6-periodic sequence {2, 1, -1, -2, -1, 1}. - Benoit Cloitre (benoit7848c(AT)orange.fr), May 23 2004

Binomial transform of 1/(1-x^3). G.f. : (1-x)^2/((1-x)^3-x^3)=x/(1-x-2x^2)+1/(1+x^3); a(n)=sum{k=0..floor(n/3), binomial(n, 3k)}; a(n)=sum{k=0..n, binomial(n, k)(cos(2*pi*k/3+pi/3)/3+sin(2*pi*k/3+pi/3)/sqrt(3)+1/3)}; a(n)=A001045(n)+sqrt(3)cos(pi*n/3+pi/6)/3+sin(pi*n/3+pi*/6)/3+(-1)^n/3. - Paul Barry (pbarry(AT)wit.ie), Jul 25 2004

a(n)=sum{k=0..n, binomial(n, 3(n-k))} - Paul Barry (pbarry(AT)wit.ie), Aug 30 2004

G.f.: ((1-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). - Michael Somos Feb 14 2006

(PARI) a(n)=sum(i=0, n, sum(j=0, n, if(n-i-3*j, 0, n!/(i)!/(3*j)!)))

(PARI) a(n) = sum(k=0, n\3, binomial(n, 3*k)) /* Michael Somos Feb 14 2006 */

(PARI) a(n)=if(n<0, 0, ([1, 0, 1; 1, 1, 0; 0, 1, 1]^n)[1, 1]) /* Michael Somos Feb 14 2006 */

Row sums of A098172.

Cf. A024494, A094715, A094717.

Sequence in context: A134508 A091357 A129715 this_sequence A130781 A071015 A084188

Adjacent sequences: A024490 A024491 A024492 this_sequence A024494 A024495 A024496

nonn

Clark Kimberling (ck6(AT)evansville.edu)