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A120714
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Sequence produced by 7 X 7 Markov chain based on adjacency matrix of 7-vertex graph with 10 edges, derived from the Fano plane.
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2
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0, 14, 42, 232, 974, 4522, 20180, 91422, 411782, 1858856, 8384078, 37827386, 170648724, 769875718, 3473203086, 15669055544, 70689396502, 318908566562, 1438725432052, 6490672907694, 29282051536966, 132103184740456
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OFFSET
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1,2
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COMMENTS
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Take the standard 7-vertex 7-edge Fano plane graph and add three edges that go around the triangle vertices from the middle of the sides ( connecting the middle of the sides without going through the center)
Characteristic polynomial is 6 - 2 x - 24 x^2 - 3 x^3 + 26 x^4 + 15 x^5 - x^7.
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LINKS
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Table of n, a(n) for n=1..22.
Eric Weisstein's World of Mathematics, Fano Plane
Index entries for linear recurrences with constant coefficients, signature (0, 15, 26, -3, -24, -2, 6).
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FORMULA
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a(n)=the first entry of the vector v(n), where v(1)=transpose(0,1,1,2,3,5,8) and v(n)=Mv(n-1) for n >=2, where M = {{0, 1, 0, 0, 0, 1, 1}, {1, 0, 1, 1, 0, 1, 1}, {0, 1, 0, 1, 0, 0, 1}, {0, 1, 1, 0, 1, 1, 1}, {0, 0, 0, 1, 0, 1, 1}, {1, 1, 0, 1, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 0}}.
Recurrence (via the Cayley-Hamilton theorem): a(n)=15a(n-2)+26a(n-3)-3a(n-4)-24a(n-5)-2a(n-6)+6a(n-7) (see the 2nd Maple program).
O.g.f.: 2*(4*x^2+14*x+7)*x^2/((-1-x+x^2)*(6*x^3+10*x^2+2*x-1)). - R. J. Mathar, Dec 05 2007
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MAPLE
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with(linalg): M := matrix(7, 7, [0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0]): v[1] := matrix(7, 1, [0, 1, 1, 2, 3, 5, 8]): for n from 2 to 25 do v[n] := multiply(M, v[n-1]) end do: seq(v[n][1, 1], n = 1 .. 25);
a[1]:=0: a[2]:=14: a[3]:=42: a[4]:=232: a[5]:=974: a[6]:=4522: a[7]:=20180: a[8]:=91422: for n from 9 to 25 do a[n]:=15*a[n-2]+26*a[n-3]-3*a[n-4]-24*a[n-5]-2*a[n-6]+6*a[n-7] end do: seq(a[n], n=1..25);
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MATHEMATICA
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M = {{0, 1, 0, 0, 0, 1, 1}, {1, 0, 1, 1, 0, 1, 1}, {0, 1, 0, 1, 0, 0, 1}, {0, 1, 1, 0, 1, 1, 1}, {0, 0, 0, 1, 0, 1, 1}, {1, 1, 0, 1, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 0}} v[1] = {0, 1, 1, 2, 3, 5, 8} v[n_] := v[n] = M.v[n - 1] a = Table[v[n][[1]], {n, 1, 50}]
LinearRecurrence[{0, 15, 26, -3, -24, -2, 6}, {0, 14, 42, 232, 974, 4522, 20180}, 30] (* Harvey P. Dale, Sep 20 2011 *)
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CROSSREFS
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Cf. A111384, A120715.
Sequence in context: A212514 A292051 A242897 * A041378 A302219 A302665
Adjacent sequences: A120711 A120712 A120713 * A120715 A120716 A120717
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KEYWORD
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nonn
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AUTHOR
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Roger L. Bagula, Aug 12 2006
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EXTENSIONS
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Edited by N. J. A. Sloane, Jul 14 2007, Jul 28 2007
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STATUS
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approved
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