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A101879
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a(0) = 1, a(1) = 1, a(2) = 2, for n>2 a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3).
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1
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1, 1, 2, 6, 21, 77, 286, 1066, 3977, 14841, 55386, 206702, 771421, 2878981, 10744502, 40099026, 149651601, 558507377, 2084377906, 7779004246, 29031639077, 108347552061, 404358569166, 1509086724602, 5631988329241, 21018866592361
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Consider the matrix M=[1,1,0; 1,3,1; 0,1,1]; Characteristic polynomial of M is x^3 - 5*x^2 + 5*x - 1. Use (M^n)[1,1] to define the recursion a(0) = 1, a(1) = 1, a(2) = 2, for n>2 a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3).
a(n+1)/a(n) converges to 2 + sqrt(3) as n goes to infinity, the largest root of the characteristic polynomial. a(n) = A061278(n) + 1; (M^n)[1,2] = A001353(n); (M^n)[1,3] = A061278(n-1) for n>0; all with the same recursive properties.
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FORMULA
| a(n)=A101265(n), n>0. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 30 2008]
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PROG
| (PARI) M=[1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=0, 40, print1((M^i)[1, 1], ", "))
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CROSSREFS
| Cf. A061278, A001353, A061278.
Sequence in context: A144904 A151287 A101265 * A063023 A150188 A150189
Adjacent sequences: A101876 A101877 A101878 * A101880 A101881 A101882
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KEYWORD
| nonn
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AUTHOR
| Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 28 2005
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