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A122605
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Let M = {{0, 1, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 1}, {-1, 4, 6, -10, -5, 6, 1}}; let v[1] = {1, 0, 0, 0, 0, 0, 0}; v[n] = M.v[n - 1]. Then a(n) = v[n][[1]]
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0
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1, 0, 0, 0, 0, 0, 0, -1, -1, -7, -8, -35, -44, -154, -208, -637, -910, -2548, -3808, -9996, -15504, -38760, -62015, -149225, -245135, -572010, -961125, -2186886, -3746886, -8348172, -14547183, -31842580, -56309764, -121415344, -217478888, -462925232, -838520240, -1765205473, -3228800413
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OFFSET
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1,10
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COMMENTS
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Characteristic polynomial: - 1 + 4 x + 6 x^2 - 10 x^3 - 5 x^4 + 6 x^5 + x^6 - x^7.
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REFERENCES
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P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31. ( Note to editor: I had the wrong Steinbach reference in the 8 X 8, 9 X 9, 10 X 10 cases )
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LINKS
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Table of n, a(n) for n=1..39.
Index to sequences with linear recurrences with constant coefficients, signature (1,6,-5,-10,6,4,-1).
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FORMULA
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a(0)=1, a(1)=a(2)=a(3)=a(4)=a(5)=a(6)=0, a(n)=a(n-1)+6a(n-2)- 5a(n-3)-10a(n-4)+6a(n-5)+4a(n-6)-a(n-7) [From Harvey P. Dale, May 02 2011]
G.f.: -x*(2*x-1)*(2*x^2-1)*(x^3+2*x^2-x-1)/((x-1)*(x^2+x-1)*(x^4-4*x^3-4*x^2+x+1)). [Colin Barker, Nov 08 2012]
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MATHEMATICA
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M = {{0, 1, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 1}, {-1, 4, 6, -10, -5, 6, 1}}; v[1] = {1, 0, 0, 0, 0, 0, 0}; v[n_] := v[n] = M.v[n - 1] a = Table[v[n][[1]], {n, 1, 50}]
LinearRecurrence[{1, 6, -5, -10, 6, 4, -1}, {1, 0, 0, 0, 0, 0, 0}, 60] (* From Harvey P. Dale, May 02 2011 *)
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CROSSREFS
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Cf. A066170.
Sequence in context: A042875 A154745 A048064 * A037953 A041106 A119453
Adjacent sequences: A122602 A122603 A122604 * A122606 A122607 A122608
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KEYWORD
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sign,easy
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AUTHOR
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Roger L. Bagula and Gary W. Adamson, Sep 20 2006
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EXTENSIONS
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Edited by N. J. A. Sloane, Feb 11 2007
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STATUS
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approved
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