(*Proof of the recurrence in the Glynn type (I) case*) (*The matrices A00, A01, A12, A23, A34, A40 in the proof of Theorem 4.8*) Adj[{a1_,a2_,b1_,b2_,b3_,b4_,c1_,c2_,c3_,c4_}, {A1_,A2_,B1_,B2_,B3_,B4_,C1_,C2_,C3_,C4_}]:= If[(a2==A1)&& (b2==B1)&& (b3==B2)&& (b4==B3)&& (c2==C1)&& (c3==C2)&& (c4==C3)&& MemberQ[{0,1},z1=a1+a2+b1-2B4]&& MemberQ[{0,1},z1-A2+2C4-c1],1,0] V0=Nest[Apply[Join,#]&, Table[{a1,a2,b1,b2,b3,b4,0,0,0,0},{a1,0,1},{a2,0,1}, {b1,0,1},{b2,0,1},{b3,0,1},{b4,0,1}],5] V1=Nest[Apply[Join,#]&, Table[{a1,a2,b1,b2,b3,b4,0,0,0,1},{a1,0,1},{a2,0,1}, {b1,0,1},{b2,0,1},{b3,0,1},{b4,0,1}],5] V2=Nest[Apply[Join,#]&, Table[{a1,a2,b1,b2,b3,b4,0,0,1,0},{a1,0,1},{a2,0,1}, {b1,0,1},{b2,0,1},{b3,0,1},{b4,0,1}],5] V3=Nest[Apply[Join,#]&, Table[{a1,a2,b1,b2,b3,b4,0,1,0,0},{a1,0,1},{a2,0,1}, {b1,0,1},{b2,0,1},{b3,0,1},{b4,0,1}],5] V4=Nest[Apply[Join,#]&, Table[{a1,a2,b1,b2,b3,b4,1,0,0,0},{a1,0,1},{a2,0,1}, {b1,0,1},{b2,0,1},{b3,0,1},{b4,0,1}],5] A00=Table[Adj[V0[[i]],V0[[j]]],{i,1,64},{j,1,64}] A01=Table[Adj[V0[[i]],V1[[j]]],{i,1,64},{j,1,64}] A12=Table[Adj[V1[[i]],V2[[j]]],{i,1,64},{j,1,64}]; A23=Table[Adj[V2[[i]],V3[[j]]],{i,1,64},{j,1,64}]; A34=Table[Adj[V3[[i]],V4[[j]]],{i,1,64},{j,1,64}]; A40=Table[Adj[V4[[i]],V0[[j]]],{i,1,64},{j,1,64}]; A=A00.A00 (*The trace function*) Tr[L_List]:=Sum[L[[k,k]],{k,1,Length[L]}] (*The recurrence*) g1[n_] := g1[n] = g1[n - 1] + g1[n - 2] + g1[n - 3] + g1[n - 4] - 1 g1[1] = 1; g1[2] = 3; g1[3] = 7; g1[4] = 13; (*The Test function: Execute Test in your Mathematica session, this verifies the recurrence (4.22) for d=13,15,...,167*) Test:=Timing[ (X=A01.A12.A23.A34.A40;For[s=1,s<=80,s++,Print["d=",2s+5,",",X=A.X;Tr[X]==g1[s+1]]]) ] Print[""]; Print["Now enter"]; Print[""]; Print["Test"]; Print[""]; Print["at your Mathematica prompt. This verifies the recurrence (4.22)"]; Print["for d=13,15,...,167."];