Fact: For the following set of entries, {[1, 2], [2, 3], [3, 4], [4, 5], [5, 1]} and the matrix A:= [ 7 10 6 2 4] [ ] [ 6 5 1 8 5] [ ] [10 2 2 4 8] [ ] [ 3 9 10 2 8] [ ] [10 9 1 6 7] Then for every integer m we have the relation m m m 133471257 (A )[1, 2] + 108644488 (A )[2, 3] - 120123406 (A )[3, 4] m m - 170198708 (A )[4, 5] + 39872623 (A )[5, 1] = 0 in Maple notation 133471257*(A^m)[1,2]+108644488*(A^m)[2,3]-120123406*(A^m)[3,4]-170198708*(A^m)[ 4,5]+39872623*(A^m)[5,1] = 0 --------------------------- ------------------------------------ Fact: For the following set of entries, {[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 1]} and the matrix A:= [ 7 3 3 5 3 8] [ ] [10 3 1 1 3 5] [ ] [ 5 10 10 10 2 8] [ ] [ 4 2 9 5 9 7] [ ] [ 4 3 10 9 10 4] [ ] [ 9 7 8 9 1 10] Then for every integer m we have the relation m m m -4686054462880 (A )[1, 2] + 1537180681477 (A )[2, 3] + 132834473938 (A )[3, 4] m m + 184921605419 (A )[4, 5] + 405282399835 (A )[5, 6] m + 878579324408 (A )[6, 1] = 0 in Maple notation -4686054462880*(A^m)[1,2]+1537180681477*(A^m)[2,3]+132834473938*(A^m)[3,4]+ 184921605419*(A^m)[4,5]+405282399835*(A^m)[5,6]+878579324408*(A^m)[6,1] = 0 --------------------------- ------------------------------------ Fact: For the following set of entries, {[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 1]} and the matrix A:= [ 2 4 7 8 1 6 4] [ ] [ 1 7 2 9 5 9 9] [ ] [ 4 3 4 6 4 4 10] [ ] [ 8 2 7 2 5 3 5] [ ] [ 7 5 4 6 8 10 2] [ ] [ 6 8 4 8 7 9 4] [ ] [10 3 2 9 1 1 5] Then for every integer m we have the relation m m 5631052003485504741 (A )[1, 2] + 466063960190162755 (A )[2, 3] m m + 2707478351015573609 (A )[3, 4] - 1240626767057032006 (A )[4, 5] m m - 2355730213201369077 (A )[5, 6] - 2785923581317481852 (A )[6, 7] m + 120292425215299208 (A )[7, 1] = 0 in Maple notation 5631052003485504741*(A^m)[1,2]+466063960190162755*(A^m)[2,3]+ 2707478351015573609*(A^m)[3,4]-1240626767057032006*(A^m)[4,5]-\ 2355730213201369077*(A^m)[5,6]-2785923581317481852*(A^m)[6,7]+ 120292425215299208*(A^m)[7,1] = 0 --------------------------- ------------------------------------ Fact: For the following set of entries, {[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 1]} and the matrix A:= [ 2 4 4 1 6 3 1 7] [ ] [ 2 1 5 10 3 10 4 6] [ ] [ 8 8 7 9 6 1 2 9] [ ] [ 5 3 2 5 3 9 8 5] [ ] [ 4 3 3 9 3 8 7 9] [ ] [10 1 5 9 1 1 3 9] [ ] [ 9 9 4 9 2 5 5 8] [ ] [ 1 2 9 5 8 2 1 9] Then for every integer m we have the relation m m 2273236442698756617349162 (A )[1, 2] + 1999819430368040394363113 (A )[2, 3] m + 767165305189814142031840 (A )[3, 4] m - 1588136850309883116048570 (A )[4, 5] m - 3959582694506026956463886 (A )[5, 6] m + 2218233441458628431907451 (A )[6, 7] m + 726071142174465563829693 (A )[7, 8] m - 2018728024137300526001872 (A )[8, 1] = 0 in Maple notation 2273236442698756617349162*(A^m)[1,2]+1999819430368040394363113*(A^m)[2,3]+ 767165305189814142031840*(A^m)[3,4]-1588136850309883116048570*(A^m)[4,5]-\ 3959582694506026956463886*(A^m)[5,6]+2218233441458628431907451*(A^m)[6,7]+ 726071142174465563829693*(A^m)[7,8]-2018728024137300526001872*(A^m)[8,1] = 0 --------------------------- ------------------------------------ Fact: For the following set of entries, {[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 1]} and the matrix A:= [ 7 1 9 5 1 4 8 5 1] [ ] [ 5 10 5 5 5 10 3 6 6] [ ] [ 7 1 4 2 8 2 2 4 1] [ ] [ 2 5 6 10 5 5 1 2 8] [ ] [10 2 5 1 3 9 2 2 3] [ ] [ 5 4 4 5 8 7 1 2 4] [ ] [ 2 2 3 2 6 4 10 4 4] [ ] [ 7 5 1 5 6 3 8 9 9] [ ] [ 5 3 1 9 8 10 7 3 7] Then for every integer m we have the relation m 1061615584564877549697124171602408 (A )[1, 2] m - 154818770040463161517799364931667 (A )[2, 3] m - 518715912702432797661737052748122 (A )[3, 4] m - 340116816750786788109642733992124 (A )[4, 5] m + 30574884876356978762653056099445 (A )[5, 6] m + 287660164139029257086433432461752 (A )[6, 7] m - 499815043026507332816825638257447 (A )[7, 8] m + 575389268148818019547545629800378 (A )[8, 9] m - 258316638892667423369471724803516 (A )[9, 1] = 0 in Maple notation 1061615584564877549697124171602408*(A^m)[1,2]-154818770040463161517799364931667 *(A^m)[2,3]-518715912702432797661737052748122*(A^m)[3,4]-\ 340116816750786788109642733992124*(A^m)[4,5]+30574884876356978762653056099445*( A^m)[5,6]+287660164139029257086433432461752*(A^m)[6,7]-\ 499815043026507332816825638257447*(A^m)[7,8]+575389268148818019547545629800378* (A^m)[8,9]-258316638892667423369471724803516*(A^m)[9,1] = 0 --------------------------- ------------------------------------ Fact: For the following set of entries, {[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 1]} and the matrix A:= [ 3 8 4 1 7 1 6 8 4 1] [ ] [ 4 10 10 3 2 6 10 1 6 7] [ ] [ 1 2 9 8 10 1 10 7 7 5] [ ] [10 3 5 5 3 4 9 5 7 5] [ ] [ 1 6 10 9 1 3 5 10 3 5] [ ] [ 9 5 1 3 2 8 6 2 8 5] [ ] [ 9 1 2 10 10 9 6 6 10 9] [ ] [ 4 9 8 8 7 6 8 9 8 1] [ ] [ 4 8 1 6 3 10 9 4 2 8] [ ] [ 2 2 7 9 6 7 7 2 2 7] Then for every integer m we have the relation m 50824708671067091278342925085862233488539919 (A )[1, 2] m - 182427060716717013174843091043191744952562997 (A )[2, 3] m + 71930492730630762178290520710993680491228648 (A )[3, 4] m - 2898571885452161493188651538443338617255227 (A )[4, 5] m + 84257070132550928864436064226027642526235727 (A )[5, 6] m + 42587905588488288449995695038569244434680222 (A )[6, 7] m + 94423135111039165052325358029670191535662378 (A )[7, 8] m + 17830446748159451400835148601006720563576671 (A )[8, 9] m + 25632692716717487495847329722940640113729156 (A )[9, 10] m - 285808929351944615102882519503349137639981141 (A )[10, 1] = 0 in Maple notation 50824708671067091278342925085862233488539919*(A^m)[1,2]-\ 182427060716717013174843091043191744952562997*(A^m)[2,3]+ 71930492730630762178290520710993680491228648*(A^m)[3,4]-\ 2898571885452161493188651538443338617255227*(A^m)[4,5]+ 84257070132550928864436064226027642526235727*(A^m)[5,6]+ 42587905588488288449995695038569244434680222*(A^m)[6,7]+ 94423135111039165052325358029670191535662378*(A^m)[7,8]+ 17830446748159451400835148601006720563576671*(A^m)[8,9]+ 25632692716717487495847329722940640113729156*(A^m)[9,10]-\ 285808929351944615102882519503349137639981141*(A^m)[10,1] = 0 --------------------------- -------------------------- This took, 0.253, seconds .