`` Let, F([m[1], m[2]]), be the number of ways of of walking from [0, 0] to the point , m = [m[1], m[2]] in the , 2, -dimensional cubic lattice using the following allowed steps: {[0, 1], [1, 0], [0, 2], [2, 0], [0, 3], [3, 0]} F(m[1], m[2]), satisfies the following linear recurrences equation with polynomial coefficients in the, m[1], direction, it satisfies 27 3 --- (m[2] + 5 + m[1]) (m[2] + 4 + m[1]) (m[2] + 3 + m[1]) (864 m[1] 44 2 2 2 + 10584 m[1] + 1080 m[2] m[1] + 39954 m[1] + 498 m[1] m[2] 2 3 + 8580 m[2] m[1] + 1911 m[2] + 83 m[2] + 15837 m[2] + 46969) F(m[1], m[2])/((m[1] + 3) (m[1] + 6) (m[1] + 9) %1) - 9/44 6 (m[2] + 5 + m[1]) (m[2] + 4 + m[1]) (33520344 + 7776 m[1] + 18362558 m[2] 3 2 3 3 + 55289243 m[1] + 329778 m[2] + 3714150 m[2] + 3735 m[1] m[2] 5 4 2 4 2 5 + 14904 m[2] m[1] + 10962 m[1] m[2] + 498 m[2] m[1] + 206712 m[1] 2 3 4 4 + 36307074 m[1] + 12191985 m[1] + 2213946 m[1] + 4980 m[2] m[1] 3 2 3 4 3 2 + 52302 m[2] m[1] + 232105 m[1] m[2] + 11122 m[2] + 198567 m[1] m[2] 2 2 2 4 + 1299408 m[2] m[1] + 3645633 m[1] m[2] + 331884 m[1] m[2] 3 2 + 2854881 m[2] m[1] + 11846838 m[2] m[1] + 23723655 m[2] m[1]) F(m[1] + 1, m[2])/((m[1] + 3) (m[1] + 4) (m[1] + 6) (m[1] + 7) (m[1] + 9) 7 8 9 %1) - 3/44 (m[2] + 5 + m[1]) (39196980 m[1] + 2048976 m[1] + 46656 m[1] 6 4 5 6 3 + 428110002 m[1] + 1743 m[1] m[2] + 63018 m[1] m[2] 3 2 + 21243591960 m[2] + 70105888044 m[1] + 777613968 m[2] + 5790238944 m[2] 5 3 3 + 30730320720 + 1042646 m[1] m[2] + 155704800 m[1] m[2] 5 4 2 4 2 + 750461424 m[2] m[1] + 497600220 m[1] m[2] + 22180484 m[2] m[1] 5 4 7 8 + 16434 m[1] m[2] + 4738176 m[1] m[2] + 120528 m[1] m[2] 7 2 3 4 5 2 + 122796 m[1] m[2] + 24010282 m[2] m[1] + 305689 m[2] m[1] 4 4 5 3 5 3 + 432804 m[1] m[2] + 1928160 m[1] m[2] + 38346 m[2] m[1] 6 2 2 5 6 + 4284198 m[1] m[2] + 62654460 m[2] m[1] + 79770798 m[1] m[2] 4 3 5 2 + 4440926 m[2] m[1] + 2937366690 m[1] + 68620363438 m[1] 3 4 4 + 37952412870 m[1] + 13106246024 m[1] + 53921672 m[2] m[1] 3 2 3 4 + 554596172 m[2] m[1] + 1029050640 m[1] m[2] + 51042000 m[2] 3 2 2 2 2 + 2316639180 m[1] m[2] + 6319869086 m[2] m[1] + 9351716716 m[1] m[2] 4 3 2 + 4310458383 m[1] m[2] + 15461817958 m[2] m[1] + 33788036203 m[2] m[1] 5 + 41077752250 m[2] m[1] + 1280856 m[2] ) F(m[1] + 2, m[2])/((m[1] + 9) %1 (m[1] + 8) (m[1] + 7) (m[1] + 5) (m[1] + 6) (m[1] + 4) (m[1] + 3)) - 1/22 ( 7 6 8 9 37755114 m[1] + 365532 m[2] + 1875096 m[1] + 40608 m[1] 6 4 5 6 3 + 433911641 m[1] + 32656836600 + 16185 m[1] m[2] + 134275 m[1] m[2] 3 + 28518507420 m[2] + 80392972290 m[1] + 2090598168 m[2] 2 5 3 3 + 10539710460 m[2] + 10681839 m[1] m[2] + 371132785 m[1] m[2] 5 4 2 4 2 + 910231149 m[2] m[1] + 797488284 m[1] m[2] + 92130054 m[2] m[1] 5 4 6 2 7 + 62355 m[1] m[2] + 32868 m[2] m[1] + 5200080 m[1] m[2] 6 3 8 7 2 + 1826 m[2] m[1] + 125928 m[1] m[2] + 171798 m[1] m[2] 3 4 6 5 2 + 54929890 m[2] m[1] + 192394 m[2] m[1] + 2977089 m[2] m[1] 4 4 5 3 5 3 + 1679034 m[1] m[2] + 4247823 m[1] m[2] + 362007 m[2] m[1] 6 2 2 5 6 + 6262647 m[1] m[2] + 95881191 m[2] m[1] + 92059725 m[1] m[2] 4 3 5 2 + 17754577 m[2] m[1] + 3125719362 m[1] + 80033681849 m[1] 3 4 4 + 43601195766 m[1] + 14564944754 m[1] + 234455084 m[2] m[1] 3 2 3 4 + 1379316019 m[2] m[1] + 2667395480 m[1] m[2] + 233935536 m[2] 3 2 2 2 + 3881332587 m[1] m[2] + 11014804845 m[2] m[1] 2 4 3 + 16795670808 m[1] m[2] + 5478965181 m[1] m[2] + 20466370652 m[2] m[1] 2 5 + 46040685756 m[2] m[1] + 56497777609 m[2] m[1] + 14129100 m[2] ) F(m[1] + 3, m[2])/((m[1] + 4) (m[1] + 6) (m[1] + 5) (m[1] + 7) (m[1] + 8) 7 6 8 (m[1] + 9) %1) - 1/44 (-1040256 m[1] + 72708 m[2] - 24192 m[1] 6 3 - 19378488 m[1] - 9739091460 m[2] - 21723438510 m[1] - 260285208 m[2] 2 5 - 2531073420 m[2] - 13614337800 + 839925 m[1] m[2] 3 3 5 4 2 - 11521210 m[1] m[2] - 54005064 m[2] m[1] - 44078766 m[1] m[2] 4 2 6 2 7 + 1116746 m[2] m[1] + 1743 m[2] m[1] - 90720 m[1] m[2] 3 4 6 5 2 - 922450 m[2] m[1] + 22659 m[2] m[1] + 158904 m[2] m[1] 4 4 5 3 5 3 + 12388 m[1] m[2] - 28760 m[1] m[2] + 9711 m[2] m[1] 6 2 2 5 6 - 97320 m[1] m[2] - 3225720 m[2] m[1] - 3400320 m[1] m[2] 4 3 5 2 + 208118 m[2] m[1] - 204761416 m[1] - 14728908130 m[1] 3 4 4 - 5643314138 m[1] - 1345767950 m[1] + 1631224 m[2] m[1] 3 2 3 4 - 70762454 m[2] m[1] - 215229438 m[1] m[2] - 1616496 m[2] 3 2 2 2 2 - 319043124 m[1] m[2] - 1294953975 m[2] m[1] - 2803102125 m[1] m[2] 4 3 2 - 472391398 m[1] m[2] - 2464815245 m[2] m[1] - 7688017826 m[2] m[1] 5 - 13272004567 m[2] m[1] + 1429020 m[2] ) F(m[1] + 4, m[2])/((m[1] + 6) 7 (m[1] + 5) (m[1] + 7) (m[1] + 8) (m[1] + 9) %1) - 1/44 (-176256 m[1] 6 6 + 3486 m[2] - 6695136 m[1] - 6989650746 m[2] - 17942092350 m[1] 3 2 5 - 235971768 m[2] - 1846312398 m[2] - 55515 m[1] m[2] 3 3 5 4 2 - 5664678 m[1] m[2] - 15641208 m[2] m[1] - 13639104 m[1] m[2] 4 2 3 4 6 - 1076590 m[2] m[1] - 242148 m[2] m[1] + 498 m[2] m[1] 5 2 2 5 6 - 3237 m[2] m[1] - 490392 m[2] m[1] - 479520 m[1] m[2] 4 3 5 2 - 55770 m[2] m[1] - 10374407280 - 105954624 m[1] - 12316271478 m[1] 3 4 4 - 4409858808 m[1] - 900692304 m[1] - 6779540 m[2] m[1] 3 2 3 4 - 48408438 m[2] m[1] - 178109172 m[1] m[2] - 13884672 m[2] 3 2 2 2 2 - 147532686 m[1] m[2] - 771803172 m[2] m[1] - 1937953344 m[1] m[2] 4 3 2 - 206501004 m[1] m[2] - 1405051866 m[2] m[1] - 5159650637 m[2] m[1] 5 - 9603021841 m[2] m[1] - 227502 m[2] ) F(m[1] + 5, m[2])/((m[1] + 6) 6 (m[1] + 7) (m[1] + 8) (m[1] + 9) %1) - 1/44 (-222525570 + 83 m[2] 6 3 - 44064 m[1] - 192490759 m[2] - 529524987 m[1] - 8838974 m[2] 2 5 3 3 5 - 59653749 m[2] - 2739 m[1] m[2] - 67899 m[1] m[2] - 93960 m[2] m[1] 4 2 4 2 5 2 - 107694 m[1] m[2] - 22077 m[2] m[1] - 1418472 m[1] - 368210538 m[1] 3 4 4 - 114661191 m[1] - 18086958 m[1] - 258013 m[2] m[1] 3 2 3 4 - 1093839 m[2] m[1] - 5556828 m[1] m[2] - 680060 m[2] 3 2 2 2 2 - 2298021 m[1] m[2] - 17518929 m[2] m[1] - 55558920 m[1] m[2] 4 3 2 - 2563164 m[1] m[2] - 26448081 m[2] m[1] - 126396105 m[2] m[1] 5 - 269929361 m[2] m[1] - 22491 m[2] ) F(m[1] + 6, m[2])/((m[1] + 7) 5 4 (m[1] + 8) (m[1] + 9) %1) + 1/44 (-14688 m[1] - 35640 m[1] m[2] 4 3 2 3 3 - 379512 m[1] - 19698 m[1] m[2] - 3769890 m[1] - 727188 m[2] m[1] 2 2 2 2 - 5432433 m[2] m[1] - 376245 m[2] m[1] - 18247751 m[1] 3 2 3 + 1589 m[2] m[1] - 43306435 m[1] - 17871095 m[2] m[1] - 47417 m[1] m[2] 2 4 4 - 2196969 m[1] m[2] + 4316 m[2] m[1] + 6332 m[2] - 21936266 m[2] 2 3 5 - 4162890 m[2] - 233706 m[2] + 996 m[2] - 35667330) F(m[1] + 7, m[2])/( 4 3 (m[1] + 8) (m[1] + 9) %1) - 1/11 (24192 m[1] + 39744 m[2] m[1] 3 2 2 2 2 + 434160 m[1] + 529068 m[2] m[1] + 25824 m[2] m[1] + 2536716 m[1] 3 2 + 2004936 m[2] m[1] + 5539483 m[1] + 7802 m[1] m[2] + 231447 m[1] m[2] 3 4 2 + 36210 m[2] + 913 m[2] + 3839325 + 2112290 m[2] + 427110 m[2] ) F(m[1] + 8, m[2])/((m[1] + 9) %1) + F(m[1] + 9, m[2]) = 0 3 2 2 %1 := 864 m[1] + 7992 m[1] + 21378 m[1] + 16735 + 1080 m[2] m[1] 2 2 3 + 6420 m[2] m[1] + 8337 m[2] + 498 m[1] m[2] + 1413 m[2] + 83 m[2] in the, m[2], direction, it satisfies 27 3 --- (m[2] + 5 + m[1]) (m[2] + 4 + m[1]) (m[2] + 3 + m[1]) (83 m[1] 44 2 2 2 + 498 m[2] m[1] + 1911 m[1] + 1080 m[1] m[2] + 15837 m[1] 3 2 + 8580 m[2] m[1] + 864 m[2] + 39954 m[2] + 46969 + 10584 m[2] ) F(m[1], m[2])/((3 + m[2]) (9 + m[2]) (6 + m[2]) %1) - 9/44 6 (m[2] + 5 + m[1]) (m[2] + 4 + m[1]) (33520344 + 7776 m[2] + 55289243 m[2] 3 2 5 + 18362558 m[1] + 12191985 m[2] + 36307074 m[2] + 14904 m[1] m[2] 3 3 4 2 4 2 2 + 3735 m[1] m[2] + 498 m[1] m[2] + 10962 m[2] m[1] + 3714150 m[1] 3 4 4 3 2 + 329778 m[1] + 11122 m[1] + 331884 m[2] m[1] + 198567 m[2] m[1] 3 4 3 2 + 2854881 m[1] m[2] + 2213946 m[2] + 52302 m[1] m[2] 2 2 2 4 + 1299408 m[2] m[1] + 11846838 m[1] m[2] + 4980 m[1] m[2] 3 2 + 232105 m[2] m[1] + 3645633 m[2] m[1] + 23723655 m[2] m[1] 5 + 206712 m[2] ) F(m[1], 1 + m[2])/((3 + m[2]) (9 + m[2]) (4 + m[2]) 8 (7 + m[2]) (6 + m[2]) %1) - 3/44 (m[2] + 5 + m[1]) (2048976 m[2] 6 7 4 5 + 428110002 m[2] + 39196980 m[2] + 16434 m[1] m[2] + 70105888044 m[2] 3 2 + 21243591960 m[1] + 37952412870 m[2] + 68620363438 m[2] + 30730320720 9 5 3 3 + 46656 m[2] + 750461424 m[1] m[2] + 155704800 m[1] m[2] 5 4 2 4 2 + 1042646 m[2] m[1] + 22180484 m[1] m[2] + 497600220 m[2] m[1] 8 7 2 5 4 + 120528 m[1] m[2] + 122796 m[2] m[1] + 1743 m[1] m[2] 6 2 7 6 3 + 4284198 m[2] m[1] + 4738176 m[2] m[1] + 63018 m[2] m[1] 3 4 6 5 2 + 4440926 m[2] m[1] + 79770798 m[2] m[1] + 62654460 m[2] m[1] 4 4 5 3 5 3 + 432804 m[1] m[2] + 38346 m[1] m[2] + 1928160 m[2] m[1] 2 5 4 3 5 + 305689 m[2] m[1] + 24010282 m[2] m[1] + 1280856 m[1] 2 3 4 + 5790238944 m[1] + 777613968 m[1] + 51042000 m[1] 4 3 2 3 + 4310458383 m[2] m[1] + 2316639180 m[2] m[1] + 15461817958 m[1] m[2] 4 3 2 2 2 + 13106246024 m[2] + 554596172 m[1] m[2] + 6319869086 m[2] m[1] 2 4 3 + 33788036203 m[1] m[2] + 53921672 m[1] m[2] + 1029050640 m[2] m[1] 2 5 + 9351716716 m[2] m[1] + 41077752250 m[2] m[1] + 2937366690 m[2] ) F(m[1], 2 + m[2])/((4 + m[2]) (7 + m[2]) (6 + m[2]) %1 (8 + m[2]) 8 6 (9 + m[2]) (5 + m[2]) (3 + m[2])) - 1/22 (1875096 m[2] + 433911641 m[2] 6 7 4 5 + 365532 m[1] + 37755114 m[2] + 32656836600 + 62355 m[1] m[2] 6 3 + 1826 m[1] m[2] + 80392972290 m[2] + 28518507420 m[1] 3 2 9 + 43601195766 m[2] + 80033681849 m[2] + 40608 m[2] 5 3 3 5 + 910231149 m[1] m[2] + 371132785 m[1] m[2] + 10681839 m[2] m[1] 4 2 4 2 8 + 92130054 m[1] m[2] + 797488284 m[2] m[1] + 125928 m[1] m[2] 7 2 5 4 6 2 + 171798 m[2] m[1] + 16185 m[1] m[2] + 6262647 m[2] m[1] 7 6 3 3 4 + 5200080 m[2] m[1] + 134275 m[2] m[1] + 17754577 m[2] m[1] 6 5 2 4 4 + 92059725 m[2] m[1] + 95881191 m[2] m[1] + 1679034 m[1] m[2] 5 3 5 3 6 2 + 362007 m[1] m[2] + 4247823 m[2] m[1] + 32868 m[1] m[2] 2 5 6 4 3 + 2977089 m[2] m[1] + 192394 m[1] m[2] + 54929890 m[2] m[1] 5 2 3 4 + 14129100 m[1] + 10539710460 m[1] + 2090598168 m[1] + 233935536 m[1] 4 3 2 3 + 5478965181 m[2] m[1] + 3881332587 m[2] m[1] + 20466370652 m[1] m[2] 4 3 2 2 2 + 14564944754 m[2] + 1379316019 m[1] m[2] + 11014804845 m[2] m[1] 2 4 3 + 46040685756 m[1] m[2] + 234455084 m[1] m[2] + 2667395480 m[2] m[1] 2 5 + 16795670808 m[2] m[1] + 56497777609 m[2] m[1] + 3125719362 m[2] ) F(m[1], 3 + m[2])/((5 + m[2]) (9 + m[2]) (8 + m[2]) (4 + m[2]) (7 + m[2]) 8 6 6 (6 + m[2]) %1) + 1/44 (24192 m[2] + 19378488 m[2] - 72708 m[1] 7 3 + 1040256 m[2] + 21723438510 m[2] + 9739091460 m[1] + 5643314138 m[2] 2 5 3 3 + 14728908130 m[2] + 54005064 m[1] m[2] + 11521210 m[1] m[2] 5 4 2 4 2 - 839925 m[2] m[1] - 1116746 m[1] m[2] + 44078766 m[2] m[1] 6 2 7 3 4 + 97320 m[2] m[1] + 90720 m[2] m[1] - 208118 m[2] m[1] 6 5 2 4 4 + 3400320 m[2] m[1] + 3225720 m[2] m[1] - 12388 m[1] m[2] 5 3 5 3 6 2 - 9711 m[1] m[2] + 28760 m[2] m[1] - 1743 m[1] m[2] 2 5 6 4 3 - 158904 m[2] m[1] - 22659 m[1] m[2] + 922450 m[2] m[1] + 13614337800 5 2 3 4 - 1429020 m[1] + 2531073420 m[1] + 260285208 m[1] + 1616496 m[1] 4 3 2 3 + 472391398 m[2] m[1] + 319043124 m[2] m[1] + 2464815245 m[1] m[2] 4 3 2 2 2 + 1345767950 m[2] + 70762454 m[1] m[2] + 1294953975 m[2] m[1] 2 4 3 + 7688017826 m[1] m[2] - 1631224 m[1] m[2] + 215229438 m[2] m[1] 2 5 + 2803102125 m[2] m[1] + 13272004567 m[2] m[1] + 204761416 m[2] ) F(m[1], 4 + m[2])/((5 + m[2]) (9 + m[2]) (8 + m[2]) (7 + m[2]) (6 + m[2]) 6 6 7 %1) + 1/44 (6695136 m[2] - 3486 m[1] + 176256 m[2] + 17942092350 m[2] 3 2 + 6989650746 m[1] + 4409858808 m[2] + 12316271478 m[2] 5 3 3 5 + 15641208 m[1] m[2] + 5664678 m[1] m[2] + 55515 m[2] m[1] 4 2 4 2 + 1076590 m[1] m[2] + 13639104 m[2] m[1] + 10374407280 3 4 6 5 2 + 55770 m[2] m[1] + 479520 m[2] m[1] + 490392 m[2] m[1] 2 5 6 4 3 5 + 3237 m[2] m[1] - 498 m[1] m[2] + 242148 m[2] m[1] + 227502 m[1] 2 3 4 + 1846312398 m[1] + 235971768 m[1] + 13884672 m[1] 4 3 2 3 + 206501004 m[2] m[1] + 147532686 m[2] m[1] + 1405051866 m[1] m[2] 4 3 2 2 2 + 900692304 m[2] + 48408438 m[1] m[2] + 771803172 m[2] m[1] 2 4 3 + 5159650637 m[1] m[2] + 6779540 m[1] m[2] + 178109172 m[2] m[1] 2 5 + 1937953344 m[2] m[1] + 9603021841 m[2] m[1] + 105954624 m[2] ) F(m[1], 5 + m[2])/((9 + m[2]) (8 + m[2]) (7 + m[2]) (6 + m[2]) %1) + 1/44 ( 6 6 3 44064 m[2] - 83 m[1] + 529524987 m[2] + 192490759 m[1] + 114661191 m[2] 2 5 3 3 5 + 368210538 m[2] + 93960 m[1] m[2] + 67899 m[1] m[2] + 2739 m[2] m[1] 4 2 4 2 5 2 + 22077 m[1] m[2] + 107694 m[2] m[1] + 22491 m[1] + 59653749 m[1] 3 4 4 3 2 + 8838974 m[1] + 680060 m[1] + 2563164 m[2] m[1] + 2298021 m[2] m[1] 3 4 3 2 + 26448081 m[1] m[2] + 18086958 m[2] + 1093839 m[1] m[2] 2 2 2 4 + 17518929 m[2] m[1] + 126396105 m[1] m[2] + 258013 m[1] m[2] 3 2 + 5556828 m[2] m[1] + 55558920 m[2] m[1] + 269929361 m[2] m[1] 5 + 1418472 m[2] + 222525570) F(m[1], 6 + m[2])/((9 + m[2]) (8 + m[2]) 5 4 4 (7 + m[2]) %1) - 1/44 (14688 m[2] + 379512 m[2] + 35640 m[2] m[1] 3 3 2 3 + 727188 m[1] m[2] + 19698 m[2] m[1] + 3769890 m[2] 2 2 2 3 2 + 376245 m[2] m[1] + 18247751 m[2] - 1589 m[1] m[2] 2 3 4 + 5432433 m[1] m[2] + 47417 m[2] m[1] - 4316 m[1] m[2] 2 2 + 17871095 m[2] m[1] + 43306435 m[2] + 2196969 m[2] m[1] + 4162890 m[1] 3 4 5 + 233706 m[1] - 6332 m[1] + 21936266 m[1] + 35667330 - 996 m[1] ) 4 F(m[1], 7 + m[2])/((9 + m[2]) (8 + m[2]) %1) - 1/11 (24192 m[2] 3 3 2 2 2 + 434160 m[2] + 39744 m[1] m[2] + 25824 m[2] m[1] + 529068 m[1] m[2] 2 2 3 + 2536716 m[2] + 231447 m[2] m[1] + 5539483 m[2] + 7802 m[2] m[1] 4 2 3 + 2004936 m[2] m[1] + 913 m[1] + 3839325 + 427110 m[1] + 36210 m[1] + 2112290 m[1]) F(m[1], 8 + m[2])/((9 + m[2]) %1) + F(m[1], 9 + m[2]) = 0 3 2 2 %1 := 864 m[2] + 7992 m[2] + 21378 m[2] + 16735 + 1080 m[1] m[2] 2 2 3 + 6420 m[2] m[1] + 8337 m[1] + 498 m[2] m[1] + 1413 m[1] + 83 m[1] Let F(n) be the number of ways of walking from , [0, 0], to the point , [n, n], in the , 2, -dimensional cubic lattice using the following allowed steps: {[0, 1], [1, 0], [0, 2], [2, 0], [0, 3], [3, 0]} F(n) satisfies the following linear recurrence equation with polynomial coefficients 27 6 5 4 -- (2 n + 5) (2 n + 3) (n + 2) (2159820 n + 51729535 n + 512331524 n 11 3 2 + 2685392385 n + 7855362920 n + 12156926680 n + 7774504216) F(n)/( 8 7 (6 + n) (5 + n) (n + 4) %1) + 3/22 (2 n + 5) (6479460 n + 181106445 n 6 5 4 3 + 2194133652 n + 15089664334 n + 64633222552 n + 177138543905 n 2 + 304257591528 n + 300068437036 n + 130163480448) F(1 + n)/((6 + n) (5 + n) (n + 4) %1) + 1/11 (5983432778160 + 17487794745456 n 6 8 3 2 + 463673260868 n + 4345137875 n + 16333924627945 n + 22309033735242 n 5 4 9 7 + 2310061697294 n + 7574927896685 n + 140388300 n + 59109538215 n ) F(n + 2)/((6 + n) (5 + n) (n + 4) %1) - 1/22 (72096915341520 9 7 6 + 1386604440 n + 203853560899716 n + 603321890953 n + 4817445297817 n 3 5 4 + 180502550695072 n + 24459209769179 n + 81857544827115 n 8 2 + 43609894770 n + 252763865224698 n ) F(n + 3)/((6 + n) (5 + n) (n + 4) 8 7 6 %1) - 1/44 (1064791260 n + 29761825795 n + 358758961532 n 5 4 3 + 2435173058454 n + 10176857625056 n + 26804503561771 n 2 + 43436904106984 n + 39580613425932 n + 15520481133840) F(n + 4)/(%1 7 6 5 (6 + n) (5 + n)) - 1/11 (161986500 n + 3798721875 n + 37451883850 n 4 3 2 + 201299646204 n + 637136807624 n + 1187469520909 n + 1206358866294 n + 515066100120) F(5 + n)/((6 + n) %1) + F(6 + n) = 0 6 5 4 3 2 %1 := 2159820 n + 38770615 n + 286081149 n + 1110165239 n + 2388276859 n + 2698740654 n + 1250309880 subject to the initial conditions F(0) = 1, F(1) = 2, F(2) = 14, F(3) = 106, F(4) = 784, F(5) = 6040, F(6) = 47332 This implies, thanks to Birkhoff-Trijinski, that F(n) is asymptotically a constant times / 1/2 1/3 \n / / |(710 + 18 201 ) 76 | | | 0.31178167 |-------------------- + ---------------------- + 8/3| |1 + | | 3 1/2 1/3 | \ \ \ 3 (710 + 18 201 ) / 1/2 1/3 1/2 1/3 1/2 40363 11507 (710 + 18 201 ) 1790 (710 + 18 201 ) 201 - ------ + -------------------------- - -------------------------------- 109512 1040364 5808699 1/2 2/3 1/2 1/2 2/3\ / 12047 (710 + 18 201 ) 201 184475 (710 + 18 201 ) | | + --------------------------------- + ---------------------------|/n + | 3531688992 158135328 / \ 1/2 1/3 199413504779 124024851757 (710 + 18 201 ) ------------- - --------------------------------- 1607045671296 15266933877312 1/2 1/3 1/2 1238669359 (710 + 18 201 ) 201 + -------------------------------------- 5088977959104 1/2 2/3 75519359843 (710 + 18 201 ) - -------------------------------- 89252844205824 1/2 2/3 1/2\ / 1780343873 (710 + 18 201 ) 201 | / 2 |2772777048006815 - --------------------------------------| / n + |----------------- 386762324891904 / / \58663595184989184 1/2 1/3 826962170817275 (710 + 18 201 ) - ------------------------------------ 278652077128698624 1/2 1/3 1/2 3195038515909795 (710 + 18 201 ) 201 + -------------------------------------------- 12446459445081871872 1/2 2/3 17281316920609045 (710 + 18 201 ) - -------------------------------------- 84710231447124381696 1/2 2/3 1/2\ \ 42199993709443825 (710 + 18 201 ) 201 | / 3| / 1/2 - ---------------------------------------------| / n | / n 1891861835652444524544 / / / / which is roughly equal to n / 0.1837837373 0.01110548058 0.00309299931\ 0.31178167 8.524295193 |1. - ------------ - ------------- + -------------| | n 2 3 | \ n n / / 1/2 / n / For the record, the first 31 terms of the sequence are [1, 2, 14, 106, 784, 6040, 47332, 375196, 3001966, 24190148, 196034522, 1596030740, 13044459766, 106961525744, 879512777006, 7249483605580, 59881171431050, 495545064567260, 4107666916668414, 34099685718629264, 283454909832384416, 2359069189033880228, 19654983216988961582, 163922457650670949952, 1368364204041239714106, 11432217649265046026588, 95586499462404682067210, 799787912665769897997944, 6696423794264697091661452, 56102314151993931988315060, 470294954022689025744037630] The whole thing took, 13420.639, seconds of CPU time