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 A120714 Sequence produced by 7 X 7 Markov chain based on adjacency matrix of 7-vertex graph with 10 edges, derived from the Fano plane. 2
 0, 14, 42, 232, 974, 4522, 20180, 91422, 411782, 1858856, 8384078, 37827386, 170648724, 769875718, 3473203086, 15669055544, 70689396502, 318908566562, 1438725432052, 6490672907694, 29282051536966, 132103184740456 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS Eric Weisstein's World of Mathematics, Fano Plane FORMULA 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 MAPLE 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); MATHEMATICA 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] (* From Harvey P. Dale, Sep 20 2011 *) CROSSREFS Cf. A111384, A120715. Sequence in context: A163756 A005587 A212514 * A041378 A041380 A151990 Adjacent sequences:  A120711 A120712 A120713 * A120715 A120716 A120717 KEYWORD nonn AUTHOR Roger Bagula, Aug 12 2006 EXTENSIONS Edited by N. J. A. Sloane, Jul 14 2007, Jul 28 2007 STATUS approved

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