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{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 
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0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#####  Combined proo
fs for 6.94 and 6.99" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "a1:
= q^(j^2 + j) * gp(n+1, 2*j+1, 1) + nada;  # LHS of 6.94 = S(n)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1G,&*&)%\"qG,&*$%\"jG\"\"#\"\"\"F+
F-F--%#gpG6%,&%\"nGF-F-F-,&F+F,F-F-F-F-F-%%nadaGF-" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 15 "a2:= eG0(1,a1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#a2G,(*&)%\"qG,&*$%\"jG\"\"#\"\"\"F+F-F--%#gpG6%%\"nG
,&F+F,F-F-F-F-F-*&)F(,(F*F-F+!\"\"F1F-F--F/6%F1,$F+F,F-F-F-%%nadaGF-" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a3:= shiftup(2,a2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a3G,(*&)%\"qG,&*$%\"jG\"\"#\"\"\"F+
F-F--%#gpG6%%\"nG,&F+F,F-F-F-F-F-*&)F(,(F*F-F+F-F1F-F--F/6%F1,&F+F,F,F
-F-F-F-%%nadaGF-" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "a4:= eG1(2,a3);" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%#a4G,**&)%\"qG,&*$%\"jG\"\"#\"\"\"F+F-F--%#gpG6%%\"
nG,&F+F,F-F-F-F-F-*&)F(,(F*F-F+F-F1F-F--F/6%,&F1F-!\"\"F-F2F-F-F-*&)F(
,*F*F-F+\"\"$F1F-F,F-F--F/6%F8,&F+F,F,F-F-F-F-%%nadaGF-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a5:= shiftdown(3,a4);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#a5G,**&)%\"qG,&*$%\"jG\"\"#\"\"\"F+F-F--%#gpG
6%%\"nG,&F+F,F-F-F-F-F-*&)F(,(F*F-F+F-F1F-F--F/6%,&F1F-!\"\"F-F2F-F-F-
*&F4F--F/6%F8,$F+F,F-F-F-%%nadaGF-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 86 "###### Thus we have the recurrence:\n###### S(n) = S(
n-1) + q^n * S(n-2) + q^n * T(n-2)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "b1:= q^(j^2 + j) * gp(n+1, 2
*j,1) + nada;    #LHS of 6.99 = T(n)" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#b1G,&*&)%\"qG,&*$%\"jG\"\"#\"\"\"F+F-F--%#gpG6%,&%\"nGF-F-F-,$F+F
,F-F-F-%%nadaGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b2:= eG
0(1,b1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b2G,(*&)%\"qG,&*$%\"jG
\"\"#\"\"\"F+F-F--%#gpG6%%\"nG,$F+F,F-F-F-*&)F(,*F*F-F+!\"\"F1F-F-F-F-
-F/6%F1,&F+F,F6F-F-F-F-%%nadaGF-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "b3:= shiftup(2,b2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#b3G,(*&)%\"qG,&*$%\"jG\"\"#\"\"\"F+F-F--%#gpG6%%\"nG,$F+F,F-F-F-*
&)F(,*F*F-F+F-F-F-F1F-F--F/6%F1,&F+F,F-F-F-F-F-%%nadaGF-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "### So we have the recurrence:\n###
 T(n) = T(n-1) + q^(n+1) * S(n-1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "c1:= q
^(15*j^2 + 2*j       ) * Trb(n+1,6*j  ,6*j  ,1)\n    -q^(15*j^2 +22*j \+
+ 8   ) * Trb(n+1,6*j+4,6*j+4,1)\n    -q^(15*j^2 + 2*j       ) * Trb(n
  ,6*j  ,6*j  ,1)\n    +q^(15*j^2 +22*j + 8   ) * Trb(n  ,6*j+4,6*j+4,
1)\n    -q^(15*j^2 + 4*j + n +1) * Trb(n  ,6*j+1,6*j+1,1)\n    +q^(15*
j^2 +16*j + n +5) * Trb(n  ,6*j+3,6*j+3,1);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#c1G,.*&)%\"qG,&*$%\"jG\"\"#\"#:F+F,\"\"\"-%$TrbG6&,&
%\"nGF.F.F.,$F+\"\"'F4F.F.F.*&)F(,(F*F-F+\"#A\"\")F.F.-F06&F2,&F+F5\"
\"%F.F=F.F.!\"\"*&F'F.-F06&F3F4F4F.F.F?*&F7F.-F06&F3F=F=F.F.F.*&)F(,*F
*F-F+F>F3F.F.F.F.-F06&F3,&F+F5F.F.FKF.F.F?*&)F(,*F*F-F+\"#;F3F.\"\"&F.
F.-F06&F3,&F+F5\"\"$F.FSF.F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "c2:= eTrb28(1,c1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#c2G,.
*&)%\"qG,(*$%\"jG\"\"#\"#:F+F,%\"nG\"\"\"F/-%$TrbG6&F.,$F+\"\"',&F+F4F
/F/F/F/F/*&)F(,*F*F-F+!\"%F.F/F/F/F/-F16&F.,&F+F4!\"\"F/F<F/F/F/*&)F(,
(F*F-F+\"#A\"\")F/F/-F16&,&F.F/F/F/,&F+F4\"\"%F/FFF/F/F=*&F?F/-F16&F.F
FFFF/F/F/*&)F(,*F*F-F+FGF.F/F/F/F/-F16&F.F5F5F/F/F=*&)F(,*F*F-F+\"#;F.
F/\"\"&F/F/-F16&F.,&F+F4\"\"$F/FWF/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "c3:= eTrb28(3,c2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#>%#c3G,**&)%\"qG,(*$%\"jG\"\"#\"#:F+F,%\"nG\"\"\"F/-%$TrbG6&F.,$F+\"
\"',&F+F4F/F/F/F/F/*&)F(,*F*F-F+!\"%F.F/F/F/F/-F16&F.,&F+F4!\"\"F/F<F/
F/F/*&)F(,*F*F-F+\"#A\"\")F/F.F/F/-F16&F.,&F+F4\"\"%F/,&F+F4\"\"&F/F/F
/F=*&)F(,*F*F-F+FFF.F/F/F/F/-F16&F.F5F5F/F/F=" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 18 "c4:= negate(2,c3);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%#c4G,**&)%\"qG,(*$%\"jG\"\"#\"#:F+F,%\"nG\"\"\"F/-%$TrbG6&F.,$
F+\"\"',&F+F4F/F/F/F/F/*&)F(,*F*F-F+\"\"%F.F/F/F/F/-F16&F.,&F+!\"'!\"
\"F/F<F/F/F/*&)F(,*F*F-F+\"#A\"\")F/F.F/F/-F16&F.,&F+F4F9F/,&F+F4\"\"&
F/F/F/F>*&F7F/-F16&F.F5F5F/F/F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "### Terms 2 and 4 cancel by symmetry" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "c5:= c4 - op(2,c4) - op(4,c4);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#c5G,&*&)%\"qG,(*$%\"jG\"\"#\"#:F+F,%\"nG\"\"\"F/-
%$TrbG6&F.,$F+\"\"',&F+F4F/F/F/F/F/*&)F(,*F*F-F+\"#A\"\")F/F.F/F/-F16&
F.,&F+F4\"\"%F/,&F+F4\"\"&F/F/F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "c6:= shiftdown(2,c5);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#c6G,&*&)%\"qG,(*$%\"jG\"\"#\"#:F+F,%\"nG\"\"\"F/-%$TrbG6&F.,$
F+\"\"',&F+F4F/F/F/F/F/*&)F(,*F*F-F+!\")F/F/F.F/F/-F16&F.,&F+F4!\"#F/,
&F+F4!\"\"F/F/F/F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "c7:=n
egate(2,c6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c7G,&*&)%\"qG,(*$%
\"jG\"\"#\"#:F+F,%\"nG\"\"\"F/-%$TrbG6&F.,$F+\"\"',&F+F4F/F/F/F/F/*&)F
(,*F*F-F+\"\")F/F/F.F/F/-F16&F.,&F+!\"'!\"#F/,&F+F=!\"\"F/F/F/F@" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "expand( 15*j^2 + 8*j + 1 + n
 + (6*j+1)^2 + (6*j+1)*(-6*j-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
(*$%\"jG\"\"#\"#:F%F&%\"nG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "#### by Lattice Gases (2.15) they sum to 0." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " simp
lify(add(subs(n=6,c7),j=-3..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
101 "###### Now the second of the required recurrences:\n###### S(n) =
 S(n-1) + q^n * S(n-2) + q^n * T(n-2)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 456 "d1:= q^(15*j^2 +  4*j      ) * Trb(n+1, 6*j+1, 6*j+1
, 1)\n    -q^(15*j^2 + 16*j + 4  ) * Trb(n+1, 6*j+3, 6*j+3, 1)\n    -q
^(15*j^2 +  4*j      ) * Trb(n  , 6*j+1, 6*j+1, 1)\n    +q^(15*j^2 + 1
6*j + 4  ) * Trb(n  , 6*j+3, 6*j+3, 1)\n    -q^(15*j^2 +  4*j + n  ) *
 Trb(n-1, 6*j+1, 6*j+1, 1)\n    +q^(15*j^2 + 16*j + 4+n) * Trb(n-1, 6*
j+3, 6*j+3, 1)\n    -q^(15*j^2 +  2*j + n  ) * Trb(n-1, 6*j  , 6*j,   \+
1)\n    +q^(15*j^2 + 22*j + 8+n) * Trb(n-1, 6*j+4, 6*j+4, 1);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#d1G,2*&)%\"qG,&*$%\"jG\"\"#\"#:F+\"
\"%\"\"\"-%$TrbG6&,&%\"nGF/F/F/,&F+\"\"'F/F/F5F/F/F/*&)F(,(F*F-F+\"#;F
.F/F/-F16&F3,&F+F6\"\"$F/F=F/F/!\"\"*&F'F/-F16&F4F5F5F/F/F?*&F8F/-F16&
F4F=F=F/F/F/*&)F(,(F*F-F+F.F4F/F/-F16&,&F4F/F?F/F5F5F/F/F?*&)F(,*F*F-F
+F:F.F/F4F/F/-F16&FKF=F=F/F/F/*&)F(,(F*F-F+F,F4F/F/-F16&FK,$F+F6FVF/F/
F?*&)F(,*F*F-F+\"#A\"\")F/F4F/F/-F16&FK,&F+F6F.F/FhnF/F/F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "d2:= eTrb28(1,d1);" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%#d2G,2*&)%\"qG,(*$%\"jG\"\"#\"#:F+\"\"%%\"nG\"\"
\"F0-%$TrbG6&F/,&F+\"\"'F0F0,&F+F5F,F0F0F0F0*&)F(,(F*F-F+!\"#F/F0F0-F2
6&F/,$F+F5F=F0F0F0*&)F(,(F*F-F+\"#;F.F0F0-F26&,&F/F0F0F0,&F+F5\"\"$F0F
EF0F0!\"\"*&F?F0-F26&F/FEFEF0F0F0*&F'F0-F26&,&F/F0FGF0F4F4F0F0FG*&)F(,
*F*F-F+FAF.F0F/F0F0-F26&FNFEFEF0F0F0*&)F(,(F*F-F+F,F/F0F0-F26&FNF=F=F0
F0FG*&)F(,*F*F-F+\"#A\"\")F0F/F0F0-F26&FN,&F+F5F.F0FjnF0F0F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "d3:= eTrb28(3,d2);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%#d3G,2*&)%\"qG,(*$%\"jG\"\"#\"#:F+\"\"%%\"
nG\"\"\"F0-%$TrbG6&F/,&F+\"\"'F0F0,&F+F5F,F0F0F0F0*&)F(,(F*F-F+!\"#F/F
0F0-F26&F/,$F+F5F=F0F0F0*&)F(,*F*F-F+\"#;F.F0F/F0F0-F26&F/,&F+F5\"\"$F
0,&F+F5F.F0F0F0!\"\"*&)F(,*F*F-F+\"#5F/F0F,F0F0-F26&F/F6F6F0F0FG*&F'F0
-F26&,&F/F0FGF0F4F4F0F0FG*&F?F0-F26&FQFDFDF0F0F0*&)F(,(F*F-F+F,F/F0F0-
F26&FQF=F=F0F0FG*&)F(,*F*F-F+\"#A\"\")F0F/F0F0-F26&FQFFFFF0F0F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "d4:= eTrb1(3,d3);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%#d4G,2*&)%\"qG,(*$%\"jG\"\"#\"#:F+\"\"%%\"
nG\"\"\"F0-%$TrbG6&F/,&F+\"\"'F0F0,&F+F5F,F0F0F0F0*&)F(,(F*F-F+!\"#F/F
0F0-F26&F/,$F+F5F=F0F0F0*&)F(,*F*F-F+\"#;\"\"$F0F/F,F0-F26&,&F/F0!\"\"
F0,&F+F5FBF0,&F+F5F.F0F0F0FF*&)F(,*F*F-F+\"#A\"\")F0F/F0F0-F26&FE,&F+F
5\"\"&F0FPF0F0FF*&)F(,*F*F-F+\"#5F/F0F,F0F0-F26&F/F6F6F0F0FF*&F'F0-F26
&FEF4F4F0F0FF*&)F(,(F*F-F+F,F/F0F0-F26&FEF=F=F0F0FF*&FJF0-F26&FEFHFHF0
F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "d5:= eTrb00(8,d4);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#d5G,.*&)%\"qG,(*$%\"jG\"\"#\"#:
F+\"\"%%\"nG\"\"\"F0-%$TrbG6&F/,&F+\"\"'F0F0,&F+F5F,F0F0F0F0*&)F(,(F*F
-F+!\"#F/F0F0-F26&F/,$F+F5F=F0F0F0*&)F(,*F*F-F+\"#5F/F0F,F0F0-F26&F/F6
F6F0F0!\"\"*&F'F0-F26&,&F/F0FDF0F4F4F0F0FD*&)F(,(F*F-F+F,F/F0F0-F26&FH
F=F=F0F0FD*&)F(,*F*F-F+\"#A\"\"(F0F/F,F0-F26&FH,&F+F5F.F0,&F+F5\"\"&F0
F0F0FD" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "d6:= eTrb1(1,d5);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#d6G,.*&)%\"qG,**$%\"jG\"\"#\"#:
F+\"\"%%\"nGF,!\"\"\"\"\"F1-%$TrbG6&,&F/F1F0F1,&F+\"\"'F1F1,&F+F7F,F1F
1F1F1*&)F(,*F*F-F+\"#5F/F1F,F1F1-F36&F5,&F+F7\"\"$F1F?F1F1F1*&)F(,(F*F
-F+!\"#F/F1F1-F36&F/,$F+F7FGF1F1F1*&F:F1-F36&F/F8F8F1F1F0*&)F(,(F*F-F+
F,F/F1F1-F36&F5FGFGF1F1F0*&)F(,*F*F-F+\"#A\"\"(F1F/F,F1-F36&F5,&F+F7F.
F1,&F+F7\"\"&F1F1F1F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "d7
:= eTrb0(4,d6);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#d7G,**&)%\"qG,(*
$%\"jG\"\"#\"#:F+!\"#%\"nG\"\"\"F0-%$TrbG6&F/,$F+\"\"'F4F0F0F0*&)F(,(F
*F-F+\"\"%F/F,F0-F26&,&F/F0!\"\"F0,&F+F5F0F0F>F0F0F=*&)F(,(F*F-F+F,F/F
0F0-F26&F<F4F4F0F0F=*&)F(,*F*F-F+\"#A\"\"(F0F/F,F0-F26&F<,&F+F5F9F0,&F
+F5\"\"&F0F0F0F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "d8:= eT
rb29(1,d7);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#d8G,**&)%\"qG,(*$%\"
jG\"\"#\"#:F+!\"#%\"nG\"\"\"F0-%$TrbG6&,&F/F0!\"\"F0,$F+\"\"'F6F0F0F0*
&)F(,(F*F-F+!\")F/F,F0-F26&F4,&F+F7F.F0,&F+F7F5F0F0F0F0*&)F(,(F*F-F+F,
F/F0F0F1F0F5*&)F(,*F*F-F+\"#A\"\"(F0F/F,F0-F26&F4,&F+F7\"\"%F0,&F+F7\"
\"&F0F0F0F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "d9:= negate(
1,d8);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#d9G,**&)%\"qG,(*$%\"jG\"
\"#\"#:F+F,%\"nG\"\"\"F/-%$TrbG6&,&F.F/!\"\"F/,$F+!\"'F5F/F/F/*&)F(,(F
*F-F+!\")F.F,F/-F16&F3,&F+\"\"'!\"#F/,&F+F>F4F/F/F/F/*&F'F/-F16&F3,$F+
F>FDF/F/F4*&)F(,*F*F-F+\"#A\"\"(F/F.F,F/-F16&F3,&F+F>\"\"%F/,&F+F>\"\"
&F/F/F/F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "### Using symm
etry, 1st and 3rd terms cancel." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "d10:= d9 - op(1,d9) -op(3,d9);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$d10G,&*&)%\"qG,(*$%\"jG\"\"#\"#:F+!\")%\"nGF,\"\"\"-%$TrbG6&,
&F/F0!\"\"F0,&F+\"\"'!\"#F0,&F+F7F5F0F0F0F0*&)F(,*F*F-F+\"#A\"\"(F0F/F
,F0-F26&F4,&F+F7\"\"%F0,&F+F7\"\"&F0F0F0F5" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "d11:= shiftdown(2,d10);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$d11G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplify(add(subs(n=7,
d10),j=-3..3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 }