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Search: id:A122186
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| A122186 |
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First row sum of the 4 X 4 matrix M^n, where M={{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2, 1}}. |
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+0
1
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| 1, 30, 707, 16886, 403104, 9623140, 229729153, 5484227157, 130922641160, 3125460977225, 74612811302754, 1781200165693270, 42521840081752984, 1015105948653689061, 24233196047277585233, 578508865448619225434
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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These matrices resemble Hankel matrices.
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REFERENCES
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Peter Steinbach, "Golden Fields: A Case for the Heptagon", Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
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FORMULA
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a(n)=23a(n-1)+21a(n-2)-4a(n-3)-a(n-4) for n>=4; a(0)=1, a(1)=30, a(2)=707, a(3)=16886 (follows from the minimal polynomial x^4-23x^3-21x^2+4x+1 of the matrix M).
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MAPLE
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with(linalg): M[1]:=matrix(4, 4, [10, 9, 7, 4, 9, 8, 6, 3, 7, 6, 4, 2, 4, 3, 2, 1]): for n from 2 to 15 do M[n]:=multiply(M[1], M[n-1]) od: 1, seq(M[n][1, 1]+M[n][1, 2]+M[n][1, 3]+M[n][1, 4], n=1..15);
a[0]:=1: a[1]:=30: a[2]:=707: a[3]:=16886: for n from 4 to 15 do a[n]:=23*a[n-1]+21*a[n-2]-4*a[n-3]-a[n-4] od: seq(a[n], n=0..15);
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MATHEMATICA
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M = {{10, 9, 7, 4}, {9, 8, 6, 3}, {7, 6, 4, 2}, {4, 3, 2, 1}}; v[1] = {1, 1, 1, 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. A015448.
Sequence in context: A152499 A027475 A035520 this_sequence A166840 A166828 A053509
Adjacent sequences: A122183 A122184 A122185 this_sequence A122187 A122188 A122189
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KEYWORD
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nonn
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AUTHOR
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Gary Adamson and Roger Bagula (qntmpkt(AT)yahoo.com), Oct 17 2006
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 07 2006
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