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A095263
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a(n+3) = 3*a(n+2) - 2*a(n+1) + a(n).
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11
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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; internal format)
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OFFSET
| 1,2
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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 (pbarry(AT)wit.ie), 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 (rlbagulatftn(AT)yahoo.com), 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,...). [From Milan R. Janjic (agnus(AT)blic.net), Sep 24 2010]
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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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.
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
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. [From Milan R. Janjic (agnus(AT)blic.net), May 02 2010]
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EXAMPLE
| a(9) = 1081 = 3*465 - 2*200 + 86.
M^9 * [1 0 0] = [a(7) a(8) a(9)] = [200 465 1081].
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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}] (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]
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CROSSREFS
| Cf. A000931, A034943.
Cf. A010912.
Cf. A097550, A137531, A052921.
Sequence in context: A124671 A188626 A123392 * A010912 A192665 A052967
Adjacent sequences: A095260 A095261 A095262 * A095264 A095265 A095266
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), May 31 2004
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EXTENSIONS
| 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
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