1,2

a(n+1) = number of n-tuples over {0,1,2} without consecutive digits. For the general case see A096261.

Diagonal sums of Riordan array (1/(1-x)^3, x/(1-x^3)), A127893. - Paul Barry (pbarry(AT)wit.ie), Jan 07 2008

Let M = the 3 X 3 matrix [0 1 0 / 0 0 1 / 1 -2 3]; then M^n *[1 0 0] = [a(n-2) a(n-1) a(n)].

a(n)/a(n-1) tends to 2.3247179572..., an eigenvalue of M and a root of the characteristic polynomial.

Related to the Padovan sequence A000931 as follows : a(n)=A000931(3n+4). Also the binomial transform of A000931(n+4).

a(n)=sum{k=0..floor((n+1)/2), binomial(n+k, n-2k+1)}; a(n)=sum{k=0..floor((n+1)/2), binomial(n+k, 3k-1)}. - Paul Barry (pbarry(AT)wit.ie), Jul 06 2004

G.f.: 1/(1-3x+2x^2-x^3); a(n)=sum{k=0..floor(n/2), C(n+k+2,3k+2)}=sum{k=0..n, C(n,k)*sum{j=0..floor((k+4)/2), C(j,k-2j+4)}}. - Paul Barry (pbarry(AT)wit.ie), Jan 07 2008

a(9) = 1081 = 3*465 - 2*200 + 86.

M^9 * [1 0 0] = [a(7) a(8) a(9)] = [200 465 1081].

a[1] = 1; a[2] = 3; a[3] = 7; a[n_] := a[n] = 3a[n - 1] - 2a[n - 2] + a[n - 3]; Table[ a[n], {n, 22}] (* Or *)

a[n_] := (MatrixPower[{{0, 1, 2, 3}, {1, 2, 3, 0}, {2, 3, 0, 1}, {3, 0, 1, 2}}, n].{{1}, {0}, {0}, {0}})[[2, 1]]; Table[ a[n], {n, 22}] (from Robert G. Wilson v Jun 16 2004)

a=0; b=0; c=1; lst={}; Do[AppendTo[lst, a+=b]; b+=c; c+=a, {n, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 20 2009]

Cf. A000931, A034943.

Cf. A010912. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 04 2008]

Cf. A097550, A137531, A052921 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 20 2009]

Sequence in context: A124671 A123392 A130691 this_sequence A010912 A052967 A152090

Adjacent sequences: A095260 A095261 A095262 this_sequence A095264 A095265 A095266

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), May 31 2004

Edited by Paul Barry (PBARRY(AT)wit.ie), Jul 06 2004

Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 05 2004