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Search: id:A123942
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| A123942 |
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The (1,4)-entry in the 4 X 4 matrix M^n, where M={{3, 2, 1, 1}, {2, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}} (n>=0). |
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+0
2
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| 0, 1, 3, 15, 71, 340, 1626, 7778, 37205, 177966, 851280, 4072001, 19477953, 93170570, 445670811, 2131815570, 10197297001, 48777608903, 233322137235, 1116069871981, 5338593130960, 25536552265626, 122151189577128
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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Martin H. Gutknecht and Lloyd N. Trefethen, Real Polynomial Chebyshev Approximation by the Caratheodory-Fejer Method, http://links.jstor.org/sici?sici=0036-1429(198204)19%3A2%3C358%3ARPCABT%3E2.0.CO%3
Rosenblum and Rovnyak, Hardy Classes and Operator Theory, Dover, New York, 1985, page 26
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FORMULA
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a(n)=4a(n-1)+4a(n-2)-a(n-3)-a(n-4) for n>=4 (follows from the minimal polynomial of the matrix M).
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MAPLE
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with(linalg): M[1]:=matrix(4, 4, [3, 2, 1, 1, 2, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]): for n from 2 to 25 do M[n]:=multiply(M[1], M[n-1]) od: 0, seq(M[n][1, 4], n=1..25);
a[0]:=0: a[1]:=1: a[2]:=3: a[3]:=15: for n from 4 to 25 do a[n]:=4*a[n-1]+4*a[n-2]-a[n-3]-a[n-4] od: seq(a[n], n=0..25);
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MATHEMATICA
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M = {{3, 2, 1, 1}, {2, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 0}}; v[1] = {0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1] a1 = Table[v[n][[1]], {n, 1, 50}]
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CROSSREFS
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Cf. A122099, A122100.
Sequence in context: A110211 A033876 A009174 this_sequence A137638 A145839 A055837
Adjacent sequences: A123939 A123940 A123941 this_sequence A123943 A123944 A123945
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KEYWORD
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nonn
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AUTHOR
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Roger Bagula and Gary Adamson (rlbagulatftn(AT)yahoo.com), Oct 25 2006
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EXTENSIONS
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Edited by njas, Dec 04 2006
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