1,2

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.

Eric Weisstein's World of Mathematics, Fano Plane

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 (mathar(AT)strw.leidenuniv.nl), Dec 05 2007

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);

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}]

Cf. A111384, A120715.

Sequence in context: A118237 A163756 A005587 this_sequence A041378 A041380 A151990

Adjacent sequences: A120711 A120712 A120713 this_sequence A120715 A120716 A120717

nonn

Roger Bagula (rlbagulatftn(AT)yahoo.com), Aug 12 2006

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 14 2007, Jul 28 2007