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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1739 "\n\nx(n) : horizon
tal edge joining nth and n+1st vertices in upper row\nz(n) : horizonta
l edge joining nth and n+1st vertices in lower row\ny(n) : vertical ed
ge joining vertices in nth column\n\nF(n) : sum of weights of configur
ations on 2-by-n grid\nG(n) : sum of weights of configurations on 2-by
-n grid \n       with upper right vertex removed\nH(n) : sum of weight
s of configurations on 2-by-n grid \n       with lower right vertex re
moved\n\nF(n) = F(n-1) + y(n) F(n-1) + x(n-1) G(n-1) + z(n-1) H(n-1) +
 x(n-1) z(n-1) \nF(n-2)\nG(n) = F(n-1) + z(n-1) H(n-1)\nH(n) = F(n-1) \+
+ x(n-1) G(n-1)\n\nF(0) = 1\nF(1) = 1 + y(1)\nG(1) = 1\nH(1) = 1\n\nF \+
:= proc(n) option remember; if n=0 then 1; elif n=1 then 1+y(1); else \+
\nsimplify( F(n-1) + y(n)*F(n-1) + x(n-1)*G(n-1) + z(n-1)*H(n-1) + \nx
(n-1)*z(n-1)*F(n-2) );\nfi; end;\n\nG := proc(n) option remember; if n
=1 then 1; else \nsimplify( F(n-1) + b(n)*H(n-1) );\nfi; end;\n\nH := \+
proc(n) option remember; if n=1 then 1; else \nsimplify( F(n-1) + a(n)
*G(n-1) );\nfi; end;\n\nF(n) = (F(n+2) - F(n+1) - c(n+2) F(n+1) - a(n+
2) G(n+1) - b(n+2) \nH(n+1)) / (a(n+2) b(n+2))\nH(n) = (G(n+1) - F(n))
 / b(n+1)\nG(n) = (H(n+1) - F(n)) / a(n+1)\n\nF := proc(n) option reme
mber; if n=0 then 1; elif n=1 then 1+c(1); elif \nn>1 then\nsimplify( \+
F(n-1) + c(n)*F(n-1) + a(n)*G(n-1) + b(n)*H(n-1) + \na(n)*b(n)*F(n-2) \+
);\nelse simplify( (F(n+2) - F(n+1) - c(n+2)*F(n+1) - a(n+2)*G(n+1) \n
                       - b(n+2)*H(n+1)) / a(n+2) / b(n+2) );\nfi; end;
\n\nG := proc(n) option remember; if n=1 then 1; elif n>1 then \nsimpl
ify( F(n-1) + b(n)*H(n-1) );\nelse simplify( (H(n+1) - F(n)) / a(n+1) \+
);\nfi; end;\n\nH := proc(n) option remember; if n=1 then 1; elif n>1 \+
then \nsimplify( F(n-1) + a(n)*G(n-1) );\nelse simplify( (G(n+1) - F(n
)) / b(n+1) );\nfi; end;\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 371 "F := proc(n) option remember; \n\nG := proc(k) option remembe
r; if k=1 then 1; else \nsimplify( F(k-1) + z(k-1)*H(k-1) );\nfi; end;
\n\nH := proc(l) option remember; if l=1 then 1; else \nsimplify( F(l-
1) + x(l-1)*G(l-1) );\nfi; end;\n\n\nif n=0 then 1; elif n=1 then 1+y(
1); else \nsimplify( F(n-1) + y(n)*F(n-1) + x(n-1)*G(n-1) + z(n-1)*H(n
-1) + \nx(n-1)*z(n-1)*F(n-2) );\nfi; end;\n\n\n" }}{PARA 7 "" 1 "" 
{TEXT -1 41 "Warning, `G` is implicitly declared local" }}{PARA 7 "" 
1 "" {TEXT -1 41 "Warning, `H` is implicitly declared local" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%\"FGR6#%\"nG6$%\"GG%\"HG6#%)rememberG6\"C%
>8$R6#%\"kGF-F+F-@%/9$\"\"\"F7-%)simplifyG6#,&-F$6#,&F6F7!\"\"F7F7*&-%
\"zGF=F7-T#F=F7F7F-F-6$F*8%>FFR6#%\"lGF-F+F-@%F5F7-F96#,&F<F7*&-%\"xGF
=F7FC\"\"\"F7F-F-6$F)F0@'/F6\"\"!F7F5,&F7F7-%\"yG6#F7F7-F96#,,F<F7*&-F
Y6#F6F7F<F7F7*&FPFR-F0F=F7F7*&FAFR-FFF=F7F7*(FPFRFAFR-F$6#,&F6F7!\"#F7
F7F7F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,0\"\"\"F$-%\"yG6#F$F$-F&6#\"\"#F$*&F(
F$F%F$F$-%\"xGF'F$-%\"zGF'F$*&F,F$F.F$F$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "F(3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,N\"\"\"F$-%
\"yG6#F$F$-F&6#\"\"#F$*&F(F$F%F$F$-%\"xGF'F$-%\"zGF'F$*&F,F$F.F$F$-F&6
#\"\"$F$*&F1F$F%\"\"\"F$*&F1F5F(F5F$*(F1F5F(F5F%F5F$*&F1F5F,F5F$*&F1F5
F.F5F$*(F1F5F,F5F.F5F$-F-F)F$*&F;F$F%F5F$*&F;F5F.F5F$-F/F)F$*&F>F$F%F5
F$*&F>F5F,F5F$*&F;F5F>F5F$*(F;F5F>F5F%F5F$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 749 "F(n) = (F(n+2) - F(n+1) - c(n+2) F(n+1) - a(n+2) G
(n+1) - b(n+2) \nH(n+1)) / (a(n+2) b(n+2))\nH(n) = (G(n+1) - F(n)) / b
(n+1)\nG(n) = (H(n+1) - F(n)) / a(n+1)\n\nF := proc(n) option remember
; if n=0 then 1; elif n=1 then 1+c(1); elif \nn>1 then\nsimplify( F(n-
1) + c(n)*F(n-1) + a(n)*G(n-1) + b(n)*H(n-1) + \na(n)*b(n)*F(n-2) );\n
else simplify( (F(n+2) - F(n+1) - c(n+2)*F(n+1) - a(n+2)*G(n+1) \n    \+
                   - b(n+2)*H(n+1)) / a(n+2) / b(n+2) );\nfi; end;\n\n
G := proc(n) option remember; if n=1 then 1; elif n>1 then \nsimplify(
 F(n-1) + b(n)*H(n-1) );\nelse simplify( (H(n+1) - F(n)) / a(n+1) );\n
fi; end;\n\nH := proc(n) option remember; if n=1 then 1; elif n>1 then
 \nsimplify( F(n-1) + a(n)*G(n-1) );\nelse simplify( (G(n+1) - F(n)) /
 b(n+1) );\nfi; end;" }}}}{MARK "4 0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 }