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 A095263 a(n+3) = 3*a(n+2) - 2*a(n+1) + a(n). 19
 1, 3, 7, 16, 37, 86, 200, 465, 1081, 2513, 5842, 13581, 31572, 73396, 170625, 396655, 922111, 2143648, 4983377, 11584946, 26931732, 62608681, 145547525, 338356945, 786584466, 1828587033, 4250949112, 9882257736, 22973462017, 53406819691 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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, Jan 07 2008 The signed variant (-1)^(n+1)*a(n+1) is the bottom right entry of the n-th power of the matrix [[0,1,0],[0,0,1],[-1,-2,-3]]. - Roger L. Bagula, Jul 01 2007 a(n) is the number of generalized compositions of n+1 when there are i^2/2-i/2 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010 Dedrickson (Section 4.1) gives a bijection between colored compositions of n, where each part k has one of binomial(k,2) colors, and 0,1,2 strings of length n-2 without sequential digits (i.e., avoiding 01 and 12). Cf. A052529. - Peter Bala, Sep 17 2013 Counts closed walks of length (n+2) from a vertex of a unidirectional triangle containing one and two loops on remaining vertices. Equivalently, the (1,1) or top left entry of A^n where A=(0,1,0;0,1,1;1,0,2) is the adjacency matrix of digraph. - David Neil McGrath, Sep 15 2014 Except for the initial 0, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^2 - S^3; see A291000.  - Clark Kimberling, Aug 24 2017 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 C. R. Dedrickson III, Compositions, Bijections, and Enumerations Thesis, Jack N. Averitt College of Graduate Studies, Georgia Southern University, 2012. Index entries for linear recurrences with constant coefficients, signature (3,-2,1) FORMULA 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. [Is that constant equal to 1 + A060006? - Michel Marcus, Oct 11 2014] [Yes, the limit is the root of the equation -1 + 2*x - 3*x^2 + x^3 = 0, after substitution x = y + 1 we have the equation for y: -1 - y + y^3 = 0, y = A060006. - Vaclav Kotesovec, Jan 27 2015] 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, Jul 06 2004 G.f.: x/(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, Jan 07 2008 If p[i]=i(i-1)/2 and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=2, a(n-1)=det A. - Milan Janjic, May 02 2010 a(n) = A000931(3*n + 4). - Michael Somos, Sep 18 2012 EXAMPLE a(9) = 1081 = 3*465 - 2*200 + 86. M^9 * [1 0 0] = [a(7) a(8) a(9)] = [200 465 1081]. x + 3*x^2 + 7*x^3 + 16*x^4 + 37*x^5 + 86*x^6 + 200*x^7 + ... MAPLE A:= gfun:-rectoproc({a(n+3)=3*a(n+2)-2*a(n+1)+a(n), a(1)=1, a(2)=3, a(3)=7}, a(n), remember): seq(A(n), n=1..100); # Robert Israel, Sep 15 2014 MATHEMATICA 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}] (* 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 (* Vladimir Joseph Stephan Orlovsky, Jan 20 2009 *) CROSSREFS Cf. A000931, A034943, A010912, A097550, A137531, A052921, A052529, A052530. Column k=3 of A277666. Sequence in context: A124671 A188626 A123392 * A010912 A192665 A052967 Adjacent sequences:  A095260 A095261 A095262 * A095264 A095265 A095266 KEYWORD nonn,easy AUTHOR Gary W. Adamson, May 31 2004 EXTENSIONS Edited by Paul Barry, Jul 06 2004 Corrected and extended by Robert G. Wilson v, Jun 05 2004 STATUS approved

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Last modified January 20 21:36 EST 2019. Contains 319336 sequences. (Running on oeis4.)