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Search: id:A122074
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| A122074 |
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a(0)=1, a(1)=6, a(n)=7a(n-1)-2a(n-2). |
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+0
1
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| 1, 6, 40, 268, 1796, 12036, 80660, 540548, 3622516, 24276516, 162690580, 1090281028, 7306586036, 48965540196, 328145609300, 2199088184708, 14737326074356, 98763106151076, 661867090908820, 4435543424059588
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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First row sum of the matrix M^n, where M is the 3 X 3 matrix {{2,2,2},{2,3,2},{2,2,3}}.
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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)=8a(n-1)-9a(n-2)+2a(n-3); a(0)=1,a(1)=6,a(2)=40 (follows from the minimal polynomial x^3-8x^2+9x-2 of M).
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EXAMPLE
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a(2)=40 because M^2={{12,14,14},{14,17,16},{14,16,17}} and 12+14+14=40.
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MAPLE
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with(linalg): M[1]:=matrix(3, 3, [2, 2, 2, 2, 3, 2, 2, 2, 3]): for n from 2 to 20 do M[n]:=multiply(M[n-1], M[1]) od: 1, seq(M[n][1, 1]+M[n][1, 2]+M[n][1, 3], n=1..20);
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MATHEMATICA
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M = {{2, 2, 2}, {2, 3, 2}, {2, 2, 3}}; v[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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Sequence in context: A065113 A052518 A135032 this_sequence A123357 A081016 A083426
Adjacent sequences: A122071 A122072 A122073 this_sequence A122075 A122076 A122077
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KEYWORD
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nonn
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AUTHOR
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Gary Adamson (qntmpkt(AT)yahoo.com), Oct 16 2006
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 29 2006 and Dec 04 2006
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