Enumerating the Number of Ways Missionaries and Cannibals can cross the rive\ r for many Infinite Families of Problems where the Boat Size and Safety \ Margin are fixed and the Excess of Missionaries to Canniabls is also fix\ ed By Shalosh B. Ekhad M missionaries and C cannibals have to cross a river. They have a boat that\ can have at most B passengers, and at no time, on either bank of the ri\ ver, and in the boat should there be a scenario where there are both missionaries and cannibals present, the excess of the number \ of missionaries to the number cannibals MUST BE AT LEAST d Let a(M,C,B,d) be the number of ways of crossing safely In this paper we will conjecture rational function expressions for the gene\ rating functions infinity ----- \ i F(r, B, d, x) = ) a(r + i, i, B, d) x / ----- i = 0 Doing Safety Margin, 0 -------------------------------------------- -------------------------------------------- doing Boat size, 2 Proposition 1 3 2 F(0, 2, 0, x) = 4 x + 4 x + x and in Maple notation 4*x^3+4*x^2+x Proposition 2 3 2 16 x + 6 x + 2 x + 1 F(1, 2, 0, x) = - ---------------------- -1 + x and in Maple notation -(16*x^3+6*x^2+2*x+1)/(-1+x) Proposition 3 3 2 68 x + 32 x + 4 x + 1 F(2, 2, 0, x) = - ----------------------- -1 + 4 x and in Maple notation -(68*x^3+32*x^2+4*x+1)/(-1+4*x) Proposition 4 3 2 372 x + 111 x + 12 x + 1 F(3, 2, 0, x) = - -------------------------- -1 + 9 x and in Maple notation -(372*x^3+111*x^2+12*x+1)/(-1+9*x) Proposition 5 2 (14 x + 1) (148 x + 27 x + 1) F(4, 2, 0, x) = - ------------------------------ -1 + 14 x and in Maple notation -(14*x+1)*(148*x^2+27*x+1)/(-1+14*x) Proposition 6 5 4 3 2 508 x - 9540 x + 8519 x + 2323 x + 125 x + 1 F(5, 2, 0, x) = - ------------------------------------------------ 2 (-1 + x) (x - 18 x + 1) and in Maple notation -(508*x^5-9540*x^4+8519*x^3+2323*x^2+125*x+1)/(-1+x)/(x^2-18*x+1) Proposition 7 5 4 3 2 11048 x + 110806 x - 34439 x - 10133 x - 353 x - 1 F(6, 2, 0, x) = - ------------------------------------------------------ 2 (3 x - 1) (2 x + 21 x - 1) and in Maple notation -(11048*x^5+110806*x^4-34439*x^3-10133*x^2-353*x-1)/(3*x-1)/(2*x^2+21*x-1) Proposition 8 F(7, 2, 0, x) = 6 5 4 3 2 4112 x - 84875 x + 793622 x - 139899 x - 43201 x - 958 x - 1 - ----------------------------------------------------------------- 4 3 2 x - 5 x + 128 x - 29 x + 1 and in Maple notation -(4112*x^6-84875*x^5+793622*x^4-139899*x^3-43201*x^2-958*x-1)/(x^4-5*x^3+128*x^ 2-29*x+1) Proposition 9 F(8, 2, 0, x) = 6 5 4 3 2 541696 x + 4739304 x - 5059060 x + 583308 x + 180284 x + 2550 x + 1 - ------------------------------------------------------------------------ 2 (8 x - 1) (12 x - 26 x + 1) and in Maple notation -(541696*x^6+4739304*x^5-5059060*x^4+583308*x^3+180284*x^2+2550*x+1)/(8*x-1)/( 12*x^2-26*x+1) Proposition 10 7 6 5 4 F(9, 2, 0, x) = - (1774224 x - 63279616 x - 54735368 x + 31754164 x 3 2 / - 2667061 x - 736742 x - 6726 x - 1) / ( / 4 3 2 4 x - 384 x + 337 x - 39 x + 1) and in Maple notation -(1774224*x^7-63279616*x^6-54735368*x^5+31754164*x^4-2667061*x^3-736742*x^2-\ 6726*x-1)/(4*x^4-384*x^3+337*x^2-39*x+1) Proposition 11 8 7 6 5 F(10, 2, 0, x) = - (53168672 x - 1471316160 x - 736566820 x - 420250404 x 4 3 2 / + 196800216 x - 13468097 x - 2954491 x - 17667 x - 1) / ((13 x - 1) / 3 2 (4 x - 76 x + 31 x - 1)) and in Maple notation -(53168672*x^8-1471316160*x^7-736566820*x^6-420250404*x^5+196800216*x^4-\ 13468097*x^3-2954491*x^2-17667*x-1)/(13*x-1)/(4*x^3-76*x^2+31*x-1) Proposition 12 10 9 8 F(11, 2, 0, x) = (106584976 x - 2784960944 x + 37575293636 x 7 6 5 4 + 26278802680 x + 3988478173 x + 2651909559 x - 1196775142 x 3 2 / + 72470163 x + 11661814 x + 46319 x + 1) / ( / 6 5 4 3 2 x - 49 x + 550 x - 2045 x + 646 x - 49 x + 1) and in Maple notation (106584976*x^10-2784960944*x^9+37575293636*x^8+26278802680*x^7+3988478173*x^6+ 2651909559*x^5-1196775142*x^4+72470163*x^3+11661814*x^2+46319*x+1)/(x^6-49*x^5+ 550*x^4-2045*x^3+646*x^2-49*x+1) Proposition 13 11 10 9 F(12, 2, 0, x) = - (1811864356 x - 24706668512 x + 693581622440 x 8 7 6 5 + 1089002935648 x + 188690786567 x + 12694238641 x + 14657224360 x 4 3 2 / - 7059602634 x + 397454310 x + 45445376 x + 121339 x + 1) / ( / 3 2 3 2 (5 x + 176 x - 36 x + 1) (x - 14 x + 18 x - 1)) and in Maple notation -(1811864356*x^11-24706668512*x^10+693581622440*x^9+1089002935648*x^8+ 188690786567*x^7+12694238641*x^6+14657224360*x^5-7059602634*x^4+397454310*x^3+ 45445376*x^2+121339*x+1)/(5*x^3+176*x^2-36*x+1)/(x^3-14*x^2+18*x-1) -------------------------------------------- doing Boat size, 3 Proposition 14 5 4 3 2 F(0, 3, 0, x) = 25 x + 32 x + 6 x + 5 x + x and in Maple notation 25*x^5+32*x^4+6*x^3+5*x^2+x Proposition 15 3 2 (x + 1) (x + x - x + 1) F(1, 3, 0, x) = - ------------------------- -1 + x and in Maple notation -(x+1)*(x^3+x^2-x+1)/(-1+x) Proposition 16 6 5 4 3 2 4 x + 16 x + 6 x - 33 x - 5 x - 7 x - 1 F(2, 3, 0, x) = - -------------------------------------------- 2 (-1 + x) and in Maple notation -(4*x^6+16*x^5+6*x^4-33*x^3-5*x^2-7*x-1)/(-1+x)^2 Proposition 17 4 3 3 x - 9 x - x - 1 F(3, 3, 0, x) = - ------------------- 2 (-1 + x) and in Maple notation -(3*x^4-9*x^3-x-1)/(-1+x)^2 Proposition 18 F(4, 3, 0, x) = 8 7 6 5 4 3 2 14 x + 92 x + 247 x + 346 x + 145 x - 159 x + 10 x - 28 x - 3 - -------------------------------------------------------------------- 2 2 (x + 2 x - 1) and in Maple notation -(14*x^8+92*x^7+247*x^6+346*x^5+145*x^4-159*x^3+10*x^2-28*x-3)/(x^2+2*x-1)^2 Proposition 19 5 4 3 2 20 x + x + 31 x - 5 x + 5 x + 1 F(5, 3, 0, x) = - ----------------------------------- 3 2 x + 3 x - 1 and in Maple notation -(20*x^5+x^4+31*x^3-5*x^2+5*x+1)/(2*x^3+3*x-1) Proposition 20 10 9 8 7 6 5 F(6, 3, 0, x) = - (62 x + 368 x + 1453 x + 3144 x + 4933 x + 3964 x 4 3 2 / 3 2 2 + 1672 x - 1060 x + 186 x - 117 x - 5) / (x + 2 x + 3 x - 1) / and in Maple notation -(62*x^10+368*x^9+1453*x^8+3144*x^7+4933*x^6+3964*x^5+1672*x^4-1060*x^3+186*x^2 -117*x-5)/(x^3+2*x^2+3*x-1)^2 Proposition 21 6 5 4 3 2 87 x - 29 x - 16 x - 134 x + 18 x - 17 x - 1 F(7, 3, 0, x) = - ------------------------------------------------- 4 3 3 x - x - 4 x + 1 and in Maple notation -(87*x^6-29*x^5-16*x^4-134*x^3+18*x^2-17*x-1)/(3*x^4-x^3-4*x+1) Proposition 22 12 11 10 9 8 F(8, 3, 0, x) = - (4160 x + 10429 x + 18087 x + 4530 x - 32084 x 7 6 5 4 3 2 - 48088 x + 3080 x + 35471 x + 16141 x - 6061 x + 1092 x - 439 x - 7 / 4 3 2 2 ) / (4 x + 2 x - 2 x - 4 x + 1) / and in Maple notation -(4160*x^12+10429*x^11+18087*x^10+4530*x^9-32084*x^8-48088*x^7+3080*x^6+35471*x ^5+16141*x^4-6061*x^3+1092*x^2-439*x-7)/(4*x^4+2*x^3-2*x^2-4*x+1)^2 Proposition 23 7 6 5 4 3 2 356 x - 687 x - 410 x + 47 x + 573 x - 53 x + 50 x + 1 F(9, 3, 0, x) = - ------------------------------------------------------------ 2 3 2 (2 x - 5 x + 1) (2 x + x - 1) and in Maple notation -(356*x^7-687*x^6-410*x^5+47*x^4+573*x^3-53*x^2+50*x+1)/(2*x^2-5*x+1)/(2*x^3+x^ 2-1) Proposition 24 14 13 12 11 10 F(10, 3, 0, x) = (7200 x - 30000 x - 266872 x - 344896 x - 606568 x 9 8 7 6 5 4 - 680893 x + 31701 x + 750840 x + 302373 x - 205201 x - 123299 x 3 2 / 2 + 32063 x - 4723 x + 1494 x + 9) / ((2 x - 1) / 3 2 2 (6 x + 7 x + 3 x - 1) ) and in Maple notation (7200*x^14-30000*x^13-266872*x^12-344896*x^11-606568*x^10-680893*x^9+31701*x^8+ 750840*x^7+302373*x^6-205201*x^5-123299*x^4+32063*x^3-4723*x^2+1494*x+9)/(2*x-1 )^2/(6*x^3+7*x^2+3*x-1)^2 Proposition 25 8 7 6 5 4 3 F(11, 3, 0, x) = - (2136 x - 2516 x + 2701 x + 3656 x + 171 x - 2402 x 2 / 6 5 4 3 2 + 133 x - 138 x - 1) / (6 x - 7 x + 14 x + 9 x + 3 x - 6 x + 1) / and in Maple notation -(2136*x^8-2516*x^7+2701*x^6+3656*x^5+171*x^4-2402*x^3+133*x^2-138*x-1)/(6*x^6-\ 7*x^5+14*x^4+9*x^3+3*x^2-6*x+1) Proposition 26 17 16 15 14 F(12, 3, 0, x) = (139167 x + 90099 x + 37816 x - 1238128 x 13 12 11 10 9 + 7900853 x + 290367 x - 82474 x - 3894656 x - 9593714 x 8 7 6 5 4 3 - 6000354 x + 5571720 x + 3951902 x - 819595 x - 816667 x + 157197 x 2 / 5 4 3 2 2 - 17063 x + 4769 x + 11) / (6 x - 19 x - 16 x - x + 6 x - 1) / and in Maple notation (139167*x^17+90099*x^16+37816*x^15-1238128*x^14+7900853*x^13+290367*x^12-82474* x^11-3894656*x^10-9593714*x^9-6000354*x^8+5571720*x^7+3951902*x^6-819595*x^5-\ 816667*x^4+157197*x^3-17063*x^2+4769*x+11)/(6*x^5-19*x^4-16*x^3-x^2+6*x-1)^2 -------------------------------------------- doing Boat size, 4 Proposition 27 6 5 4 3 2 x (206 x + 112 x + 15 x + 23 x + 4 x + 1) F(0, 4, 0, x) = - ---------------------------------------------- -1 + x and in Maple notation -x*(206*x^6+112*x^5+15*x^4+23*x^3+4*x^2+1)/(-1+x) Proposition 28 14 13 12 11 10 9 8 F(1, 4, 0, x) = - (40 x + 8 x + 3 x - 36 x - 13 x - 6 x - 76 x 7 6 5 4 3 2 / 3 - 9 x + 4 x + 92 x + 13 x - 2 x + 16 x + x + 1) / ((-1 + x) / 2 3 (x + x + 1) ) and in Maple notation -(40*x^14+8*x^13+3*x^12-36*x^11-13*x^10-6*x^9-76*x^8-9*x^7+4*x^6+92*x^5+13*x^4-\ 2*x^3+16*x^2+x+1)/(-1+x)^3/(x^2+x+1)^3 Proposition 29 15 13 12 11 10 9 F(2, 4, 0, x) = - (162 x + 27 x + 2457 x + 297 x + 27 x - 2007 x 8 7 6 5 4 3 2 / - 270 x - 18 x + 387 x + 27 x - x + 106 x + 10 x + x + 1) / / 3 3 (3 x - 1) and in Maple notation -(162*x^15+27*x^13+2457*x^12+297*x^11+27*x^10-2007*x^9-270*x^8-18*x^7+387*x^6+ 27*x^5-x^4+106*x^3+10*x^2+x+1)/(3*x^3-1)^3 Proposition 30 21 20 19 18 17 16 F(3, 4, 0, x) = - (12 x - 3 x - 360 x - 244 x + 18 x + 3450 x 15 14 13 12 11 10 9 + 1572 x + 36 x - 8292 x - 3522 x - 264 x - 315 x + 1289 x 8 7 6 5 4 3 2 / + 36 x + 2256 x + 66 x + 18 x - 717 x - 36 x - 3 x - 18 x - 1) / / 6 3 3 (x - 5 x + 1) and in Maple notation -(12*x^21-3*x^20-360*x^19-244*x^18+18*x^17+3450*x^16+1572*x^15+36*x^14-8292*x^ 13-3522*x^12-264*x^11-315*x^10+1289*x^9+36*x^8+2256*x^7+66*x^6+18*x^5-717*x^4-\ 36*x^3-3*x^2-18*x-1)/(x^6-5*x^3+1)^3 Proposition 31 23 21 20 19 18 17 F(4, 4, 0, x) = - (40 x + 3 x - 1239 x - 62 x - 93 x + 11648 x 16 15 14 13 12 11 + 904 x + 914 x - 29454 x - 2778 x - 2955 x - 41254 x 10 9 8 7 6 5 4 - 4207 x + 475 x + 16325 x + 2387 x - 47 x - 2478 x - 159 x 3 2 / 6 3 3 + 12 x - 226 x - 9 x - 1) / (x - 8 x + 1) / and in Maple notation -(40*x^23+3*x^21-1239*x^20-62*x^19-93*x^18+11648*x^17+904*x^16+914*x^15-29454*x ^14-2778*x^13-2955*x^12-41254*x^11-4207*x^10+475*x^9+16325*x^8+2387*x^7-47*x^6-\ 2478*x^5-159*x^4+12*x^3-226*x^2-9*x-1)/(x^6-8*x^3+1)^3 Proposition 32 33 30 29 28 27 26 F(5, 4, 0, x) = - (36 x + 872 x + 3 x - 2 x + 9438 x + 126 x 25 24 23 22 21 20 19 - 11 x + 60415 x + 1303 x + 13 x + 247521 x + 7172 x + 456 x 18 17 16 15 14 13 + 658708 x + 23538 x + 2139 x + 1042456 x + 44880 x + 4836 x 12 11 10 9 8 7 + 774736 x + 41559 x + 4123 x - 145177 x - 8148 x - 164 x 6 5 4 3 2 / + 2687 x - 713 x - 53 x + 1606 x + 80 x + 3 x + 6) / / 9 6 3 3 (x + 5 x + 13 x - 1) and in Maple notation -(36*x^33+872*x^30+3*x^29-2*x^28+9438*x^27+126*x^26-11*x^25+60415*x^24+1303*x^ 23+13*x^22+247521*x^21+7172*x^20+456*x^19+658708*x^18+23538*x^17+2139*x^16+ 1042456*x^15+44880*x^14+4836*x^13+774736*x^12+41559*x^11+4123*x^10-145177*x^9-\ 8148*x^8-164*x^7+2687*x^6-713*x^5-53*x^4+1606*x^3+80*x^2+3*x+6)/(x^9+5*x^6+13*x ^3-1)^3 Proposition 33 31 30 28 27 26 F(6, 4, 0, x) = - (16000 x + 10750 x + 215775 x + 137950 x + 750 x 25 24 23 22 21 + 1179100 x + 708665 x + 7575 x + 3419770 x + 1861721 x 20 19 18 17 16 + 29740 x + 5682486 x + 2629829 x + 57411 x + 5413113 x 15 14 13 12 11 + 1872456 x + 56442 x + 2659977 x + 481716 x + 25238 x 10 9 8 7 6 5 + 367714 x - 50156 x + 2706 x - 117311 x - 6539 x - 525 x 4 3 2 / 9 6 3 3 + 10237 x + 338 x + 16 x + 120 x + 3) / (5 x + 21 x + 18 x - 1) / and in Maple notation -(16000*x^31+10750*x^30+215775*x^28+137950*x^27+750*x^26+1179100*x^25+708665*x^ 24+7575*x^23+3419770*x^22+1861721*x^21+29740*x^20+5682486*x^19+2629829*x^18+ 57411*x^17+5413113*x^16+1872456*x^15+56442*x^14+2659977*x^13+481716*x^12+25238* x^11+367714*x^10-50156*x^9+2706*x^8-117311*x^7-6539*x^6-525*x^5+10237*x^4+338*x ^3+16*x^2+120*x+3)/(5*x^9+21*x^6+18*x^3-1)^3 -------------------------------------------- doing Boat size, 5 Proposition 34 F(0, 5, 0, x) = 8 7 6 5 4 3 2 x (582 x + 138 x - 104 x + 79 x + 32 x - 14 x + 15 x + 1) - ---------------------------------------------------------------- -1 + x and in Maple notation -x*(582*x^8+138*x^7-104*x^6+79*x^5+32*x^4-14*x^3+15*x^2+1)/(-1+x) Proposition 35 F(1, 5, 0, x) = 11 10 9 8 7 6 5 3 2 4 x - x - 29 x - x + 14 x + 2 x + 20 x + 10 x - x + x + 1 -------------------------------------------------------------------- 2 2 (-1 + x) (x + 1) and in Maple notation (4*x^11-x^10-29*x^9-x^8+14*x^7+2*x^6+20*x^5+10*x^3-x^2+x+1)/(-1+x)^2/(x+1)^2 Proposition 36 16 14 13 12 11 10 9 F(2, 5, 0, x) = - (4 x + 66 x + 7 x - 64 x - 17 x - 315 x - 3 x 8 7 6 5 4 3 2 / + 564 x + 26 x - 313 x - 11 x - 39 x - x - 30 x - x - 1) / ( / 4 4 (-1 + x) (x + 1) ) and in Maple notation -(4*x^16+66*x^14+7*x^13-64*x^12-17*x^11-315*x^10-3*x^9+564*x^8+26*x^7-313*x^6-\ 11*x^5-39*x^4-x^3-30*x^2-x-1)/(-1+x)^4/(x+1)^4 Proposition 37 16 15 14 13 12 11 10 F(3, 5, 0, x) = - (4 x - x + 29 x + 2 x - 199 x - 4 x + 445 x 9 8 7 6 5 4 3 2 + 9 x - 397 x - 9 x + 139 x + 4 x + 23 x - 2 x + 19 x + x + 1) / 5 5 / ((-1 + x) (x + 1) ) / and in Maple notation -(4*x^16-x^15+29*x^14+2*x^13-199*x^12-4*x^11+445*x^10+9*x^9-397*x^8-9*x^7+139*x ^6+4*x^5+23*x^4-2*x^3+19*x^2+x+1)/(-1+x)^5/(x+1)^5 Proposition 38 21 19 17 16 15 14 F(4, 5, 0, x) = - (24 x - 212 x + 667 x - 49 x - 1008 x + 226 x 13 12 11 10 9 8 7 + 1766 x - 445 x - 2758 x + 461 x + 3529 x - 246 x - 1840 x 6 5 4 3 2 / 7 + 60 x + 680 x - 13 x + 274 x + 5 x + 30 x + 1) / ((-1 + x) / 7 (x + 1) ) and in Maple notation -(24*x^21-212*x^19+667*x^17-49*x^16-1008*x^15+226*x^14+1766*x^13-445*x^12-2758* x^11+461*x^10+3529*x^9-246*x^8-1840*x^7+60*x^6+680*x^5-13*x^4+274*x^3+5*x^2+30* x+1)/(-1+x)^7/(x+1)^7 Proposition 39 22 21 20 19 18 17 F(5, 5, 0, x) = - (4 x + 35 x - 20 x - 240 x + 60 x + 694 x 16 15 14 13 12 11 10 - 119 x - 1417 x + 179 x + 2171 x - 220 x - 2712 x + 187 x 9 8 7 6 5 3 2 + 2298 x - 90 x - 801 x + 21 x + 96 x + 114 x - 3 x + 18 x + 1) / 6 2 6 6 / ((x + 2 x - 1) (-1 + x) (x + 1) ) / and in Maple notation -(4*x^22+35*x^21-20*x^20-240*x^19+60*x^18+694*x^17-119*x^16-1417*x^15+179*x^14+ 2171*x^13-220*x^12-2712*x^11+187*x^10+2298*x^9-90*x^8-801*x^7+21*x^6+96*x^5+114 *x^3-3*x^2+18*x+1)/(x^6+2*x^2-1)/(-1+x)^6/(x+1)^6 Proposition 40 32 30 28 27 26 F(6, 5, 0, x) = - (2430 x + 6480 x + 82494 x + 1134 x - 72933 x 25 24 23 22 21 20 - 1296 x + 355870 x + 5211 x - 361972 x - 5073 x + 639912 x 19 18 17 16 15 14 + 8814 x - 588269 x - 7245 x + 566601 x + 6474 x - 417927 x 13 12 11 10 9 8 - 4346 x + 237427 x + 1698 x - 118513 x - 778 x + 36013 x 7 6 5 4 3 2 / - 21 x - 7339 x + 91 x + 871 x + 18 x - 543 x - 9 x - 10) / ( / 2 4 4 4 (3 x - 1) (x + 1) ) and in Maple notation -(2430*x^32+6480*x^30+82494*x^28+1134*x^27-72933*x^26-1296*x^25+355870*x^24+ 5211*x^23-361972*x^22-5073*x^21+639912*x^20+8814*x^19-588269*x^18-7245*x^17+ 566601*x^16+6474*x^15-417927*x^14-4346*x^13+237427*x^12+1698*x^11-118513*x^10-\ 778*x^9+36013*x^8-21*x^7-7339*x^6+91*x^5+871*x^4+18*x^3-543*x^2-9*x-10)/(3*x^2-\ 1)^4/(x^4+1)^4 Proposition 41 26 25 24 23 22 F(7, 5, 0, x) = - (7305 x + 10 x - 71249 x - 128 x + 190990 x 21 20 19 18 17 16 + 355 x - 133947 x + 15 x + 271542 x + 480 x - 150818 x 15 14 13 12 11 10 9 + 332 x + 141627 x + 291 x - 71281 x + 278 x + 30787 x + 23 x 8 7 6 5 4 3 2 - 12327 x + 28 x + 3272 x - 10 x - 570 x - 13 x + 231 x + 3 x + 6) / 6 4 2 8 6 4 2 2 / ((5 x + x + 3 x - 1) (2 x - 10 x + 3 x - 4 x + 1) ) / and in Maple notation -(7305*x^26+10*x^25-71249*x^24-128*x^23+190990*x^22+355*x^21-133947*x^20+15*x^ 19+271542*x^18+480*x^17-150818*x^16+332*x^15+141627*x^14+291*x^13-71281*x^12+ 278*x^11+30787*x^10+23*x^9-12327*x^8+28*x^7+3272*x^6-10*x^5-570*x^4-13*x^3+231* x^2+3*x+6)/(5*x^6+x^4+3*x^2-1)/(2*x^8-10*x^6+3*x^4-4*x^2+1)^2 Doing Safety Margin, 1 -------------------------------------------- -------------------------------------------- doing Boat size, 2 Proposition 42 3 2 3 x - 11 x + x - 1 F(5, 2, 1, x) = - -------------------- 2 (-1 + x) and in Maple notation -(3*x^3-11*x^2+x-1)/(-1+x)^2 Proposition 43 2 x - 1 F(6, 2, 1, x) = -------- -1 + 4 x and in Maple notation (2*x-1)/(-1+4*x) Proposition 44 3 (-1 + x) F(7, 2, 1, x) = - ------------ 2 x - 6 x + 1 and in Maple notation -(-1+x)^3/(x^2-6*x+1) Proposition 45 3 2 8 x - 20 x + 4 x - 1 F(8, 2, 1, x) = - ---------------------- 2 4 x - 8 x + 1 and in Maple notation -(8*x^3-20*x^2+4*x-1)/(4*x^2-8*x+1) Proposition 46 6 5 4 3 2 16 x - 55 x + 71 x - 27 x + 73 x - 5 x + 1 F(9, 2, 1, x) = ----------------------------------------------- 4 3 2 x - 6 x + 11 x - 10 x + 1 and in Maple notation (16*x^6-55*x^5+71*x^4-27*x^3+73*x^2-5*x+1)/(x^4-6*x^3+11*x^2-10*x+1) Proposition 47 6 5 4 3 2 128 x - 314 x + 445 x - 64 x + 210 x - 6 x + 1 F(10, 2, 1, x) = --------------------------------------------------- 2 2 (x - 10 x + 1) (-1 + x) and in Maple notation (128*x^6-314*x^5+445*x^4-64*x^3+210*x^2-6*x+1)/(x^2-10*x+1)/(-1+x)^2 Proposition 48 F(11, 2, 1, x) = 7 6 5 4 3 2 648 x - 1028 x + 1064 x - 1732 x + 33 x - 543 x + 7 x - 1 - --------------------------------------------------------------- 4 3 2 4 x - 24 x + 37 x - 14 x + 1 and in Maple notation -(648*x^7-1028*x^6+1064*x^5-1732*x^4+33*x^3-543*x^2+7*x-1)/(4*x^4-24*x^3+37*x^2 -14*x+1) Proposition 49 8 7 6 5 4 3 F(12, 2, 1, x) = (8464 x - 26864 x + 21220 x - 3432 x + 5344 x + 672 x 2 / 2 2 + 1332 x - 8 x + 1) / ((6 x - 12 x + 1) (2 x - 4 x + 1)) / and in Maple notation (8464*x^8-26864*x^7+21220*x^6-3432*x^5+5344*x^4+672*x^3+1332*x^2-8*x+1)/(6*x^2-\ 12*x+1)/(2*x^2-4*x+1) -------------------------------------------- doing Boat size, 3 Proposition 50 2 F(1, 3, 1, x) = x + x + 1 and in Maple notation x^2+x+1 Proposition 51 F(2, 3, 1, x) = 1 + 2 x and in Maple notation 1+2*x Proposition 52 4 3 2 15 x - 8 x + 8 x + 1 F(3, 3, 1, x) = - ----------------------- -1 + x and in Maple notation -(15*x^4-8*x^3+8*x^2+1)/(-1+x) Proposition 53 4 3 2 13 x - 14 x + x - 6 x - 3 F(4, 3, 1, x) = - ---------------------------- 2 (-1 + x) and in Maple notation -(13*x^4-14*x^3+x^2-6*x-3)/(-1+x)^2 Proposition 54 4 3 2 4 x + 5 x - x + 3 x + 1 F(5, 3, 1, x) = - -------------------------- 2 x + x - 1 and in Maple notation -(4*x^4+5*x^3-x^2+3*x+1)/(x^2+x-1) Proposition 55 6 5 4 3 2 63 x + 260 x + 174 x - 182 x + 50 x - 28 x - 5 F(6, 3, 1, x) = - --------------------------------------------------- 2 2 (x + 2 x - 1) and in Maple notation -(63*x^6+260*x^5+174*x^4-182*x^3+50*x^2-28*x-5)/(x^2+2*x-1)^2 Proposition 56 5 4 3 2 13 x + 43 x + 30 x - 7 x + 10 x + 1 F(7, 3, 1, x) = - --------------------------------------- 3 2 x + 3 x + 2 x - 1 and in Maple notation -(13*x^5+43*x^4+30*x^3-7*x^2+10*x+1)/(x^3+3*x^2+2*x-1) Proposition 57 8 7 6 5 4 3 F(8, 3, 1, x) = - (260 x + 1262 x + 3201 x + 3552 x + 1857 x - 1354 x 2 / 3 2 2 + 349 x - 122 x - 7) / (x + 2 x + 3 x - 1) / and in Maple notation -(260*x^8+1262*x^7+3201*x^6+3552*x^5+1857*x^4-1354*x^3+349*x^2-122*x-7)/(x^3+2* x^2+3*x-1)^2 Proposition 58 6 5 4 3 2 23 x + 32 x - 229 x - 159 x + 24 x - 29 x - 1 F(9, 3, 1, x) = - -------------------------------------------------- 4 2 x - 4 x - 3 x + 1 and in Maple notation -(23*x^6+32*x^5-229*x^4-159*x^3+24*x^2-29*x-1)/(x^4-4*x^2-3*x+1) Proposition 59 10 9 8 7 6 F(10, 3, 1, x) = - (13808 x + 25032 x - 15544 x - 55660 x - 12073 x 5 4 3 2 / + 35456 x + 18155 x - 7904 x + 1656 x - 452 x - 9) / / 4 3 2 2 (4 x + 2 x - 2 x - 4 x + 1) and in Maple notation -(13808*x^10+25032*x^9-15544*x^8-55660*x^7-12073*x^6+35456*x^5+18155*x^4-7904*x ^3+1656*x^2-452*x-9)/(4*x^4+2*x^3-2*x^2-4*x+1)^2 Proposition 60 F(11, 3, 1, x) = 8 7 6 5 4 3 2 242 x - 53 x + 886 x + 819 x - 990 x - 768 x + 76 x - 80 x - 1 - --------------------------------------------------------------------- 5 4 3 2 2 x + 9 x + 5 x - 4 x - 4 x + 1 and in Maple notation -(242*x^8-53*x^7+886*x^6+819*x^5-990*x^4-768*x^3+76*x^2-80*x-1)/(2*x^5+9*x^4+5* x^3-4*x^2-4*x+1) Proposition 61 12 11 10 9 8 F(12, 3, 1, x) = - (145640 x - 67744 x + 176442 x + 740850 x + 199742 x 7 6 5 4 3 2 - 739736 x - 395626 x + 206844 x + 136869 x - 41180 x + 6595 x / 2 3 2 2 - 1506 x - 11) / ((2 x - 1) (6 x + 7 x + 3 x - 1) ) / and in Maple notation -(145640*x^12-67744*x^11+176442*x^10+740850*x^9+199742*x^8-739736*x^7-395626*x^ 6+206844*x^5+136869*x^4-41180*x^3+6595*x^2-1506*x-11)/(2*x-1)^2/(6*x^3+7*x^2+3* x-1)^2 -------------------------------------------- doing Boat size, 4 Proposition 62 3 2 F(1, 4, 1, x) = 4 x + 3 x + x + 1 and in Maple notation 4*x^3+3*x^2+x+1 Proposition 63 2 F(2, 4, 1, x) = 8 x + x + 1 and in Maple notation 8*x^2+x+1 Proposition 64 6 5 4 3 2 8 x - 22 x + 20 x + 2 x - 7 x + 7 x + 1 F(3, 4, 1, x) = - -------------------------------------------- -1 + x and in Maple notation -(8*x^6-22*x^5+20*x^4+2*x^3-7*x^2+7*x+1)/(-1+x) Proposition 65 6 5 4 3 2 2 x - 9 x + 35 x - 67 x + 13 x + x + 1 F(4, 4, 1, x) = ------------------------------------------- 2 x - 3 x + 1 and in Maple notation (2*x^6-9*x^5+35*x^4-67*x^3+13*x^2+x+1)/(x^2-3*x+1) Proposition 66 7 6 5 4 3 2 77 x - 223 x - 194 x + 367 x - 48 x - 8 x + 27 x - 6 F(5, 4, 1, x) = ---------------------------------------------------------- 3 2 (-1 + x) (x - 2 x - 4 x + 1) and in Maple notation (77*x^7-223*x^6-194*x^5+367*x^4-48*x^3-8*x^2+27*x-6)/(-1+x)/(x^3-2*x^2-4*x+1) Proposition 67 12 11 10 9 8 7 6 F(6, 4, 1, x) = - (18 x - x + 102 x - 25 x + 7 x - 349 x - 21 x 5 4 3 2 / 3 2 3 - 11 x + 284 x + 25 x + 5 x + 63 x + 3) / ((-1 + x) (x + x + 1) ) / and in Maple notation -(18*x^12-x^11+102*x^10-25*x^9+7*x^8-349*x^7-21*x^6-11*x^5+284*x^4+25*x^3+5*x^2 +63*x+3)/(-1+x)^3/(x^2+x+1)^3 Proposition 68 30 29 28 27 26 25 24 F(7, 4, 1, x) = - (6 x + 88 x + 3 x - 34 x - 610 x - 3 x + 138 x 23 22 21 20 19 18 17 + 2161 x - 32 x - 329 x - 4705 x + 120 x + 601 x + 7067 x 16 15 14 13 12 11 10 - 262 x - 684 x - 6651 x + 244 x + 564 x + 3722 x - 193 x 9 8 7 6 5 4 3 2 - 176 x - 629 x + 57 x + 22 x - 32 x - 96 x - 5 x + 257 x + 26 x / 9 6 3 3 + 1) / (x - 2 x + 4 x - 1) / and in Maple notation -(6*x^30+88*x^29+3*x^28-34*x^27-610*x^26-3*x^25+138*x^24+2161*x^23-32*x^22-329* x^21-4705*x^20+120*x^19+601*x^18+7067*x^17-262*x^16-684*x^15-6651*x^14+244*x^13 +564*x^12+3722*x^11-193*x^10-176*x^9-629*x^8+57*x^7+22*x^6-32*x^5-96*x^4-5*x^3+ 257*x^2+26*x+1)/(x^9-2*x^6+4*x^3-1)^3 Proposition 69 30 29 28 27 26 F(8, 4, 1, x) = - (389844 x + 24260 x - 100 x - 560180 x - 39304 x 25 24 23 22 21 20 + 1280 x + 1141902 x + 60212 x - 2216 x - 982908 x - 54062 x 19 18 17 16 15 14 + 3628 x + 955680 x + 37107 x - 3325 x - 486598 x - 17623 x 13 12 11 10 9 8 7 + 2782 x + 243400 x + 4277 x - 1404 x - 56208 x + 55 x + 622 x 6 5 4 3 2 / + 4041 x - 604 x - 123 x + 692 x + 82 x + 8 x + 15) / / 9 6 3 3 (10 x - 4 x + 7 x - 1) and in Maple notation -(389844*x^30+24260*x^29-100*x^28-560180*x^27-39304*x^26+1280*x^25+1141902*x^24 +60212*x^23-2216*x^22-982908*x^21-54062*x^20+3628*x^19+955680*x^18+37107*x^17-\ 3325*x^16-486598*x^15-17623*x^14+2782*x^13+243400*x^12+4277*x^11-1404*x^10-\ 56208*x^9+55*x^8+622*x^7+4041*x^6-604*x^5-123*x^4+692*x^3+82*x^2+8*x+15)/(10*x^ 9-4*x^6+7*x^3-1)^3 -------------------------------------------- doing Boat size, 5 Proposition 70 4 3 2 F(1, 5, 1, x) = 10 x + 2 x + x + x + 1 and in Maple notation 10*x^4+2*x^3+x^2+x+1 Proposition 71 3 2 F(2, 5, 1, x) = 20 x + 9 x + x + 1 and in Maple notation 20*x^3+9*x^2+x+1 Proposition 72 11 10 9 8 7 6 5 4 F(3, 5, 1, x) = - (110 x - 12 x - 129 x + 22 x + 2 x - 4 x - x - 2 x 3 2 / 2 2 + x - 3 x - x - 1) / ((-1 + x) (x + 1) ) / and in Maple notation -(110*x^11-12*x^10-129*x^9+22*x^8+2*x^7-4*x^6-x^5-2*x^4+x^3-3*x^2-x-1)/(-1+x)^2 /(x+1)^2 Proposition 73 14 13 12 11 10 9 8 F(4, 5, 1, x) = (2 x + 23 x - 14 x - 151 x + 33 x + 340 x - 31 x 7 6 5 4 3 2 / 4 - 310 x + 13 x + 70 x - 6 x + 20 x + 2 x + 17 x + 1) / ((-1 + x) / 4 (x + 1) ) and in Maple notation (2*x^14+23*x^13-14*x^12-151*x^11+33*x^10+340*x^9-31*x^8-310*x^7+13*x^6+70*x^5-6 *x^4+20*x^3+2*x^2+17*x+1)/(-1+x)^4/(x+1)^4 Proposition 74 15 14 13 12 11 10 9 8 F(5, 5, 1, x) = (x - x - 7 x - x - 24 x + 8 x + 129 x - 10 x 7 6 5 4 3 2 / 4 - 153 x + 6 x + 45 x - 2 x + 17 x - x + 10 x + 1) / ((-1 + x) / 4 (x + 1) ) and in Maple notation (x^15-x^14-7*x^13-x^12-24*x^11+8*x^10+129*x^9-10*x^8-153*x^7+6*x^6+45*x^5-2*x^4 +17*x^3-x^2+10*x+1)/(-1+x)^4/(x+1)^4 Proposition 75 18 17 16 15 14 13 F(6, 5, 1, x) = (16 x - 4 x - 316 x + 3 x + 1619 x + 44 x 12 11 10 9 8 7 6 - 3531 x - 142 x + 4070 x + 180 x - 2312 x - 97 x + 643 x 5 4 3 2 / 6 6 + 20 x - 91 x - 12 x + 180 x + 8 x + 10) / ((-1 + x) (x + 1) ) / and in Maple notation (16*x^18-4*x^17-316*x^16+3*x^15+1619*x^14+44*x^13-3531*x^12-142*x^11+4070*x^10+ 180*x^9-2312*x^8-97*x^7+643*x^6+20*x^5-91*x^4-12*x^3+180*x^2+8*x+10)/(-1+x)^6/( x+1)^6 Proposition 76 18 17 16 15 14 13 12 F(7, 5, 1, x) = - (118 x + x - 94 x + x + 456 x + 4 x - 567 x 11 10 8 7 6 5 3 2 + 3 x + 614 x - 743 x - x + 440 x - 8 x - 3 x + 94 x + 3 x + 6) / 6 2 2 / ((-1 + x) (x + 1) (x + 2 x - 1) ) / and in Maple notation -(118*x^18+x^17-94*x^16+x^15+456*x^14+4*x^13-567*x^12+3*x^11+614*x^10-743*x^8-x ^7+440*x^6-8*x^5-3*x^3+94*x^2+3*x+6)/(-1+x)/(x+1)/(x^6+2*x^2-1)^2 Proposition 77 29 28 27 26 25 F(8, 5, 1, x) = - (36 x + 3510 x + 38547 x - 4743 x - 50607 x 24 23 22 21 20 19 + 16473 x + 166493 x - 18687 x - 180934 x + 30538 x + 268434 x 18 17 16 15 14 13 - 29306 x - 229884 x + 28923 x + 188276 x - 22834 x - 118473 x 12 11 10 9 8 7 6 + 14906 x + 51127 x - 8807 x - 16498 x + 3887 x + 363 x - 1359 x 5 4 3 2 / 2 4 + 1947 x + 326 x - 184 x - 24 x - 179 x - 3) / ((3 x - 1) / 4 4 (x + 1) ) and in Maple notation -(36*x^29+3510*x^28+38547*x^27-4743*x^26-50607*x^25+16473*x^24+166493*x^23-\ 18687*x^22-180934*x^21+30538*x^20+268434*x^19-29306*x^18-229884*x^17+28923*x^16 +188276*x^15-22834*x^14-118473*x^13+14906*x^12+51127*x^11-8807*x^10-16498*x^9+ 3887*x^8+363*x^7-1359*x^6+1947*x^5+326*x^4-184*x^3-24*x^2-179*x-3)/(3*x^2-1)^4/ (x^4+1)^4 Proposition 78 25 24 23 22 21 20 F(9, 5, 1, x) = (3 x - 410 x - 8680 x + 1868 x + 39173 x - 251 x 19 18 17 16 15 14 + 2991 x + 3253 x + 52459 x - 816 x - 998 x + 2371 x 13 12 11 10 9 8 7 + 16486 x - 851 x - 2067 x + 1009 x - 1983 x - 411 x + 360 x 6 5 4 3 2 / + 197 x - 799 x - 78 x + 154 x + 6 x + 85 x + 1) / ( / 8 6 4 2 6 4 2 2 (2 x - 10 x + 3 x - 4 x + 1) (5 x + x + 3 x - 1) ) and in Maple notation (3*x^25-410*x^24-8680*x^23+1868*x^22+39173*x^21-251*x^20+2991*x^19+3253*x^18+ 52459*x^17-816*x^16-998*x^15+2371*x^14+16486*x^13-851*x^12-2067*x^11+1009*x^10-\ 1983*x^9-411*x^8+360*x^7+197*x^6-799*x^5-78*x^4+154*x^3+6*x^2+85*x+1)/(2*x^8-10 *x^6+3*x^4-4*x^2+1)/(5*x^6+x^4+3*x^2-1)^2 Doing Safety Margin, 2 -------------------------------------------- -------------------------------------------- doing Boat size, 2 Proposition 79 2 2 x - x + 1 F(7, 2, 2, x) = ------------ 2 (-1 + x) and in Maple notation (2*x^2-x+1)/(-1+x)^2 Proposition 80 2 6 x + 2 x - 1 F(8, 2, 2, x) = -------------- -1 + 4 x and in Maple notation (6*x^2+2*x-1)/(-1+4*x) Proposition 81 3 2 2 x - 12 x - 3 x + 1 F(9, 2, 2, x) = ---------------------- 2 x - 6 x + 1 and in Maple notation (2*x^3-12*x^2-3*x+1)/(x^2-6*x+1) Proposition 82 3 2 10 x - 16 x - 4 x + 1 F(10, 2, 2, x) = ----------------------- 2 4 x - 8 x + 1 and in Maple notation (10*x^3-16*x^2-4*x+1)/(4*x^2-8*x+1) Proposition 83 6 5 4 3 2 4 x - 10 x - 4 x + 30 x - 11 x - 5 x + 1 F(11, 2, 2, x) = --------------------------------------------- 4 3 2 x - 6 x + 11 x - 10 x + 1 and in Maple notation (4*x^6-10*x^5-4*x^4+30*x^3-11*x^2-5*x+1)/(x^4-6*x^3+11*x^2-10*x+1) Proposition 84 6 5 4 3 2 32 x - 74 x + 58 x + 74 x + 15 x - 6 x + 1 F(12, 2, 2, x) = ----------------------------------------------- 2 2 (x - 10 x + 1) (-1 + x) and in Maple notation (32*x^6-74*x^5+58*x^4+74*x^3+15*x^2-6*x+1)/(x^2-10*x+1)/(-1+x)^2 -------------------------------------------- doing Boat size, 3 Proposition 85 F(4, 3, 2, x) = 3 and in Maple notation 3 Proposition 86 F(6, 3, 2, x) = 5 and in Maple notation 5 Proposition 87 3 x - 1 F(7, 3, 2, x) = -------- -1 + 4 x and in Maple notation (3*x-1)/(-1+4*x) Proposition 88 6 5 4 3 2 13 x + 4 x - 81 x - 22 x - 25 x - 10 x - 7 F(8, 3, 2, x) = - ----------------------------------------------- 2 2 (-1 + x) (x + 1) and in Maple notation -(13*x^6+4*x^5-81*x^4-22*x^3-25*x^2-10*x-7)/(-1+x)^2/(x+1)^2 Proposition 89 7 6 5 4 3 2 172 x + 344 x + 15 x - 140 x + 4 x - 13 x - 2 x - 1 F(9, 3, 2, x) = --------------------------------------------------------- 2 (x + 1) (x + x - 1) and in Maple notation (172*x^7+344*x^6+15*x^5-140*x^4+4*x^3-13*x^2-2*x-1)/(x+1)/(x^2+x-1) Proposition 90 5 4 3 2 72 x + 122 x - 24 x - 78 x + 8 x + 9 F(10, 3, 2, x) = ---------------------------------------- 2 2 (x + 1) (2 x - 1) and in Maple notation (72*x^5+122*x^4-24*x^3-78*x^2+8*x+9)/(x+1)^2/(2*x-1)^2 Proposition 91 5 4 3 2 105 x + 148 x + 13 x - 31 x - 3 x - 1 F(11, 3, 2, x) = ----------------------------------------- 2 (x + 1) (3 x + x - 1) and in Maple notation (105*x^5+148*x^4+13*x^3-31*x^2-3*x-1)/(x+1)/(3*x^2+x-1) Proposition 92 9 8 7 6 5 4 3 F(12, 3, 2, x) = - (20 x + 88 x - 362 x - 1489 x - 1404 x + 6 x + 496 x 2 / 2 2 2 + 108 x - 46 x - 11) / ((x + 1) (4 x + x - 1) ) / and in Maple notation -(20*x^9+88*x^8-362*x^7-1489*x^6-1404*x^5+6*x^4+496*x^3+108*x^2-46*x-11)/(x+1)^ 2/(4*x^2+x-1)^2 -------------------------------------------- doing Boat size, 4 Proposition 93 2 F(2, 4, 2, x) = x + x + 1 and in Maple notation x^2+x+1 Proposition 94 F(3, 4, 2, x) = 1 + 2 x and in Maple notation 1+2*x Proposition 95 F(4, 4, 2, x) = 1 + 3 x and in Maple notation 1+3*x Proposition 96 3 2 F(5, 4, 2, x) = x + 16 x + x + 6 and in Maple notation x^3+16*x^2+x+6 Proposition 97 4 3 2 17 x + 40 x + 4 x + 17 x + 3 F(6, 4, 2, x) = - ------------------------------- -1 + x and in Maple notation -(17*x^4+40*x^3+4*x^2+17*x+3)/(-1+x) Proposition 98 5 4 3 2 89 x + 121 x + 28 x + 18 x + 6 x + 1 F(7, 4, 2, x) = - ---------------------------------------- 2 2 x + 2 x - 1 and in Maple notation -(89*x^5+121*x^4+28*x^3+18*x^2+6*x+1)/(2*x^2+2*x-1) Proposition 99 12 11 9 8 7 6 5 4 F(8, 4, 2, x) = - (15 x + 8 x - 44 x - 16 x + 4 x - 2 x - 4 x - 8 x 3 2 / 3 2 3 + 52 x + 12 x + 4 x + 15) / ((-1 + x) (x + x + 1) ) / and in Maple notation -(15*x^12+8*x^11-44*x^9-16*x^8+4*x^7-2*x^6-4*x^5-8*x^4+52*x^3+12*x^2+4*x+15)/(-\ 1+x)^3/(x^2+x+1)^3 Proposition 100 30 28 27 25 24 22 F(9, 4, 2, x) = - (4 x + 60 x - 41 x + 80 x + 226 x - 836 x 21 20 19 18 17 16 15 - 684 x + 4 x + 2436 x + 1423 x - 16 x - 3644 x - 1872 x 14 13 12 11 10 9 8 + 48 x + 3840 x + 1640 x - 72 x - 3036 x - 707 x + 80 x 7 6 5 4 3 2 / + 1924 x + 126 x - 32 x - 628 x - 24 x + 4 x + 112 x + 5) / / 9 6 3 3 (x - 2 x + 4 x - 1) and in Maple notation -(4*x^30+60*x^28-41*x^27+80*x^25+226*x^24-836*x^22-684*x^21+4*x^20+2436*x^19+ 1423*x^18-16*x^17-3644*x^16-1872*x^15+48*x^14+3840*x^13+1640*x^12-72*x^11-3036* x^10-707*x^9+80*x^8+1924*x^7+126*x^6-32*x^5-628*x^4-24*x^3+4*x^2+112*x+5)/(x^9-\ 2*x^6+4*x^3-1)^3 Proposition 101 32 31 30 29 28 F(10, 4, 2, x) = - (144 x - 240 x + 400 x + 94808 x + 13056 x 27 26 25 24 23 22 - 5320 x - 222288 x + 30928 x + 11824 x + 230972 x - 29896 x 21 20 19 18 17 16 - 14164 x - 299208 x + 54692 x + 15600 x + 202660 x - 42684 x 15 14 13 12 11 10 - 8926 x - 94992 x + 25688 x + 5192 x + 32456 x - 13296 x 9 8 7 6 5 4 3 2 - 1305 x - 796 x + 3788 x + 171 x - 1508 x - 604 x - 17 x + 296 x / 9 6 3 3 + 40 x + 1) / (10 x - 4 x + 7 x - 1) / and in Maple notation -(144*x^32-240*x^31+400*x^30+94808*x^29+13056*x^28-5320*x^27-222288*x^26+30928* x^25+11824*x^24+230972*x^23-29896*x^22-14164*x^21-299208*x^20+54692*x^19+15600* x^18+202660*x^17-42684*x^16-8926*x^15-94992*x^14+25688*x^13+5192*x^12+32456*x^ 11-13296*x^10-1305*x^9-796*x^8+3788*x^7+171*x^6-1508*x^5-604*x^4-17*x^3+296*x^2 +40*x+1)/(10*x^9-4*x^6+7*x^3-1)^3 -------------------------------------------- doing Boat size, 5 Proposition 102 3 2 F(2, 5, 2, x) = 4 x + 3 x + x + 1 and in Maple notation 4*x^3+3*x^2+x+1 Proposition 103 2 F(3, 5, 2, x) = 3 x + x + 1 and in Maple notation 3*x^2+x+1 Proposition 104 2 F(4, 5, 2, x) = 4 x + 7 x + 1 and in Maple notation 4*x^2+7*x+1 Proposition 105 5 4 3 2 F(5, 5, 2, x) = 4 x + 18 x + 2 x + x + 3 x + 1 and in Maple notation 4*x^5+18*x^4+2*x^3+x^2+3*x+1 Proposition 106 5 4 3 2 6 x + 9 x - 25 x + 22 x - 6 x + 10 F(6, 5, 2, x) = - -------------------------------------- -1 + x and in Maple notation -(6*x^5+9*x^4-25*x^3+22*x^2-6*x+10)/(-1+x) Proposition 107 6 5 4 3 2 (x + 1) (x + 14 x + 22 x - 54 x + 7 x - 21 x + 6) F(7, 5, 2, x) = - ------------------------------------------------------ 2 3 x + 3 x - 1 and in Maple notation -(x+1)*(x^6+14*x^5+22*x^4-54*x^3+7*x^2-21*x+6)/(3*x^2+3*x-1) Proposition 108 14 13 12 11 10 9 F(8, 5, 2, x) = - (9 x + 72 x - 2880 x + 195 x - 2529 x + 2802 x 8 7 6 5 4 3 2 / - 137 x + 2494 x - 881 x - x - 832 x + 80 x + 11 x + 93 x + 3) / / 3 3 (3 x - 1) and in Maple notation -(9*x^14+72*x^13-2880*x^12+195*x^11-2529*x^10+2802*x^9-137*x^8+2494*x^7-881*x^6 -x^5-832*x^4+80*x^3+11*x^2+93*x+3)/(3*x^3-1)^3 Proposition 109 14 13 12 11 10 9 8 F(9, 5, 2, x) = (9 x - 2 x - 1537 x - 73 x - 96 x - 104 x + 5187 x 7 6 5 4 3 2 / 2 + 14 x + 3657 x + 57 x - x + 59 x + 5 x + 49 x + 1) / ((-1 + x) / 2 2 2 (x + 1) (x + 1) ) and in Maple notation (9*x^14-2*x^13-1537*x^12-73*x^11-96*x^10-104*x^9+5187*x^8+14*x^7+3657*x^6+57*x^ 5-x^4+59*x^3+5*x^2+49*x+1)/(-1+x)^2/(x+1)^2/(x^2+1)^2 Proposition 110 24 23 22 21 20 19 F(10, 5, 2, x) = - (139 x + 12 x + 966 x + 72 x + 2343 x + 130 x 18 17 16 15 14 13 12 + 1533 x + 5 x - 2346 x - 186 x - 2957 x - 82 x + 1744 x 11 10 9 8 7 6 5 4 + 74 x + 2146 x - 40 x - 1490 x - 50 x - 736 x + 77 x + 139 x 3 2 / 2 2 4 2 4 + 26 x - 362 x - 26 x - 35) / ((x + 1) (x + x - 1) ) / and in Maple notation -(139*x^24+12*x^23+966*x^22+72*x^21+2343*x^20+130*x^19+1533*x^18+5*x^17-2346*x^ 16-186*x^15-2957*x^14-82*x^13+1744*x^12+74*x^11+2146*x^10-40*x^9-1490*x^8-50*x^ 7-736*x^6+77*x^5+139*x^4+26*x^3-362*x^2-26*x-35)/(x^2+1)^2/(x^4+x^2-1)^4 Proposition 111 28 26 24 23 22 21 F(11, 5, 2, x) = - (51 x + 374 x - 259 x - x - 7954 x - x 20 19 18 17 16 15 14 - 21666 x + 24 x - 21075 x + 33 x - 3625 x - 93 x + 9057 x 13 12 11 10 9 8 7 - 301 x + 4154 x - 285 x - 1911 x - 33 x - 1808 x + 106 x 6 5 4 3 2 / 2 - 340 x + 54 x - 228 x - 7 x - 195 x - 8 x - 15) / ((x + 1) / 6 4 2 8 6 4 2 (x - 3 x - x + 1) (x + 5 x + 3 x - 1) ) and in Maple notation -(51*x^28+374*x^26-259*x^24-x^23-7954*x^22-x^21-21666*x^20+24*x^19-21075*x^18+ 33*x^17-3625*x^16-93*x^15+9057*x^14-301*x^13+4154*x^12-285*x^11-1911*x^10-33*x^ 9-1808*x^8+106*x^7-340*x^6+54*x^5-228*x^4-7*x^3-195*x^2-8*x-15)/(x^2+1)/(x^6-3* x^4-x^2+1)/(x^8+5*x^6+3*x^4-1)^2 Doing Safety Margin, 3 -------------------------------------------- -------------------------------------------- doing Boat size, 2 Proposition 112 2 2 x - x + 1 F(9, 2, 3, x) = ------------ 2 (-1 + x) and in Maple notation (2*x^2-x+1)/(-1+x)^2 Proposition 113 2 6 x + 2 x - 1 F(10, 2, 3, x) = -------------- -1 + 4 x and in Maple notation (6*x^2+2*x-1)/(-1+4*x) Proposition 114 3 2 2 x - 12 x - 3 x + 1 F(11, 2, 3, x) = ---------------------- 2 x - 6 x + 1 and in Maple notation (2*x^3-12*x^2-3*x+1)/(x^2-6*x+1) Proposition 115 3 2 10 x - 16 x - 4 x + 1 F(12, 2, 3, x) = ----------------------- 2 4 x - 8 x + 1 and in Maple notation (10*x^3-16*x^2-4*x+1)/(4*x^2-8*x+1) -------------------------------------------- doing Boat size, 3 Proposition 116 F(4, 3, 3, x) = 3 and in Maple notation 3 Proposition 117 F(6, 3, 3, x) = 5 and in Maple notation 5 Proposition 118 F(8, 3, 3, x) = 7 and in Maple notation 7 Proposition 119 21 x + 1 F(9, 3, 3, x) = - -------- -1 + 4 x and in Maple notation -(21*x+1)/(-1+4*x) Proposition 120 8 7 6 5 4 3 2 10 x + 10 x - 37 x - 8 x + 53 x + 2 x - 35 x + 12 x + 9 F(10, 3, 3, x) = -------------------------------------------------------------- 4 4 (-1 + x) (x + 1) and in Maple notation (10*x^8+10*x^7-37*x^6-8*x^5+53*x^4+2*x^3-35*x^2+12*x+9)/(-1+x)^4/(x+1)^4 Proposition 121 F(11, 3, 3, x) = 8 7 6 5 4 3 2 275 x + 550 x + 71 x - 119 x + 28 x - 68 x - 12 x - x - 1 ---------------------------------------------------------------- 2 (x + 1) (x + x - 1) and in Maple notation (275*x^8+550*x^7+71*x^6-119*x^5+28*x^4-68*x^3-12*x^2-x-1)/(x+1)/(x^2+x-1) Proposition 122 5 4 3 2 84 x + 144 x - 24 x - 87 x + 6 x + 11 F(12, 3, 3, x) = ----------------------------------------- 2 2 (x + 1) (2 x - 1) and in Maple notation (84*x^5+144*x^4-24*x^3-87*x^2+6*x+11)/(x+1)^2/(2*x-1)^2 -------------------------------------------- doing Boat size, 4 Proposition 123 F(5, 4, 3, x) = 6 and in Maple notation 6 Proposition 124 F(6, 4, 3, x) = 3 and in Maple notation 3 Proposition 125 F(8, 4, 3, x) = 15 and in Maple notation 15 Proposition 126 11 x - 5 F(9, 4, 3, x) = -------- -1 + 4 x and in Maple notation (11*x-5)/(-1+4*x) Proposition 127 4 3 2 12 x - 16 x - 7 x + 8 x + 1 F(10, 4, 3, x) = ------------------------------ 2 2 (2 x - 1) (2 x + 1) and in Maple notation (12*x^4-16*x^3-7*x^2+8*x+1)/(2*x-1)^2/(2*x+1)^2 Proposition 128 15 14 12 11 10 9 8 F(11, 4, 3, x) = - (35 x + 4 x - 484 x - 52 x - x + 608 x + 60 x 7 6 5 4 3 2 / 4 + x + 68 x + 20 x + x - 215 x - 32 x - x - 28) / ((-1 + x) / 2 4 (x + x + 1) ) and in Maple notation -(35*x^15+4*x^14-484*x^12-52*x^11-x^10+608*x^9+60*x^8+x^7+68*x^6+20*x^5+x^4-215 *x^3-32*x^2-x-28)/(-1+x)^4/(x^2+x+1)^4 Proposition 129 12 11 10 9 8 7 F(12, 4, 3, x) = - (1394 x - 12 x - 536 x - 2060 x - 106 x + 856 x 6 5 4 3 2 / 3 3 + 856 x + 36 x - 454 x + 196 x + 10 x + 80 x + 7) / (2 x - 1) / and in Maple notation -(1394*x^12-12*x^11-536*x^10-2060*x^9-106*x^8+856*x^7+856*x^6+36*x^5-454*x^4+ 196*x^3+10*x^2+80*x+7)/(2*x^3-1)^3 -------------------------------------------- doing Boat size, 5 Proposition 130 2 F(3, 5, 3, x) = x + x + 1 and in Maple notation x^2+x+1 Proposition 131 F(4, 5, 3, x) = 2 x + 1 and in Maple notation 2*x+1 Proposition 132 F(5, 5, 3, x) = 1 + 3 x and in Maple notation 1+3*x Proposition 133 F(6, 5, 3, x) = 10 + 6 x and in Maple notation 10+6*x Proposition 134 3 2 F(7, 5, 3, x) = x + 25 x + x + 6 and in Maple notation x^3+25*x^2+x+6 Proposition 135 2 F(8, 5, 3, x) = 45 x + 32 x + 3 and in Maple notation 45*x^2+32*x+3 Proposition 136 6 5 4 3 2 37 x + 89 x + 81 x + 104 x + 32 x + 15 x + 1 F(9, 5, 3, x) = - ------------------------------------------------- 2 x + x - 1 and in Maple notation -(37*x^6+89*x^5+81*x^4+104*x^3+32*x^2+15*x+1)/(x^2+x-1) Proposition 137 12 11 9 7 6 5 F(10, 5, 3, x) = - (912 x + 192 x - 808 x + 32 x - 152 x - 176 x 4 3 2 / 3 3 - 32 x + 126 x + 64 x + 8 x + 35) / (2 x - 1) / and in Maple notation -(912*x^12+192*x^11-808*x^9+32*x^7-152*x^6-176*x^5-32*x^4+126*x^3+64*x^2+8*x+35 )/(2*x^3-1)^3 Proposition 138 15 14 12 11 10 9 8 F(11, 5, 3, x) = - (52 x + 12 x - 23 x + 44 x - 12 x + 4 x + 21 x 7 6 5 4 3 2 / 3 + 56 x - 28 x - 8 x - 13 x + 312 x + 28 x + 4 x + 15) / ((-1 + x) / 3 2 3 (x + 1) (x + 1) ) and in Maple notation -(52*x^15+12*x^14-23*x^12+44*x^11-12*x^10+4*x^9+21*x^8+56*x^7-28*x^6-8*x^5-13*x ^4+312*x^3+28*x^2+4*x+15)/(-1+x)^3/(x+1)^3/(x^2+1)^3 Proposition 139 27 25 24 23 22 21 F(12, 5, 3, x) = - (24 x + 188 x - 17 x + 280 x - 136 x - 1008 x 20 19 18 17 16 15 14 - 404 x - 3672 x - 484 x - 3432 x - 8 x + 712 x + 444 x 13 12 11 10 9 8 7 + 1352 x + 182 x - 2304 x - 184 x - 2344 x - 76 x + 296 x 6 5 4 3 2 / 2 4 + 68 x + 760 x + 24 x - 200 x - 12 x - 236 x - 5) / ((x + 1) / 4 2 4 (x + x - 1) ) and in Maple notation -(24*x^27+188*x^25-17*x^24+280*x^23-136*x^22-1008*x^21-404*x^20-3672*x^19-484*x ^18-3432*x^17-8*x^16+712*x^15+444*x^14+1352*x^13+182*x^12-2304*x^11-184*x^10-\ 2344*x^9-76*x^8+296*x^7+68*x^6+760*x^5+24*x^4-200*x^3-12*x^2-236*x-5)/(x^2+1)^4 /(x^4+x^2-1)^4 This ends this paper that took, 2781.019, seconds to generate