Displaying 1-10 of 12 results found.
Petersen graph (3,1) coloring a rectangular array: number of n X 3 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.
+10
1
9, 115, 1519, 20115, 266419, 3528715, 46737819, 619042315, 8199214219, 108598575915, 1438387920619, 19051445129515, 252336352607019, 3342194485203115, 44267359266773419, 586321084882796715
FORMULA
Empirical: a(n) = 15*a(n-1) - 24*a(n-2) + 10*a(n-3).
G.f.: x*(9 - 20*x + 10*x^2) / ((1 - x)*(1 - 14*x + 10*x^2)).
a(n) = (13 + (13-2*sqrt(39))*(7-sqrt(39))^n + (7+sqrt(39))^n*(13+2*sqrt(39))) / 39.
(End)
EXAMPLE
Some solutions for n=3:
..0..1..4....0..3..4....0..1..4....0..2..5....0..1..4....0..3..4....0..3..0
..0..3..0....4..3..5....2..1..0....0..3..0....4..3..0....4..3..4....4..3..0
..5..2..5....5..3..4....4..1..4....0..1..0....0..3..4....5..3..4....5..3..0
Petersen graph (3,1) coloring a rectangular array: number of nX4 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0
+10
1
27, 631, 16323, 426359, 11148439, 291545903, 7624417031, 199391762123, 5214442630935, 136366781617267, 3566229514618067, 93263130563653603, 2438993757290874987, 63783946691623236183, 1668061610819558039475
FORMULA
Empirical: a(n) = 31*a(n-1) -127*a(n-2) -20*a(n-3) +705*a(n-4) -1027*a(n-5) +499*a(n-6) -60*a(n-7)
EXAMPLE
Some solutions for n=3
..0..1..4..1....0..1..2..1....0..3..5..4....0..3..0..3....0..1..0..3
..4..3..4..3....4..1..4..5....5..3..5..3....4..3..4..1....0..1..4..1
..0..3..0..1....4..5..2..1....0..3..5..2....5..3..0..1....0..3..4..3
Petersen graph (3,1) coloring a rectangular array: number of nX5 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0
+10
1
81, 3539, 182901, 9685063, 515473927, 27465794119, 1463848507173, 78024299447333, 4158831849750231, 221674060909378867, 11815685765605683663, 629800688938588467995, 33569692923595929936491, 1789334831509984492336661
FORMULA
Empirical: a(n) = 80*a(n-1) -1601*a(n-2) +9025*a(n-3) +32750*a(n-4) -458870*a(n-5) +1007560*a(n-6) +2753424*a(n-7) -13680802*a(n-8) +9570798*a(n-9) +33912359*a(n-10) -66671806*a(n-11) +25819908*a(n-12) +31393403*a(n-13) -30099964*a(n-14) +2740719*a(n-15) +5650986*a(n-16) -2070082*a(n-17) -348*a(n-18) +116444*a(n-19) -20740*a(n-20) +1120*a(n-21)
EXAMPLE
Some solutions for n=3
..0..2..0..2..0....0..3..5..3..5....0..3..4..5..3....0..3..0..3..4
..0..2..5..2..5....0..3..5..2..0....0..1..4..5..3....0..3..0..1..4
..5..2..1..2..5....5..2..0..2..1....4..1..2..5..3....4..3..0..1..2
Petersen graph (3,1) coloring a rectangular array: number of nX6 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0
+10
1
243, 19759, 2030665, 216562815, 23328902821, 2519813048575, 272386213374733, 29451199763005655, 3184571844145868835, 344356382352508380215, 37236420474777196695869, 4026507614168634996035183
FORMULA
Empirical: a(n) = 171*a(n-1) -7597*a(n-2) +66978*a(n-3) +2583824*a(n-4) -51950940*a(n-5) -114768696*a(n-6) +9054636698*a(n-7) -36718682736*a(n-8) -634109162555*a(n-9) +4922415752542*a(n-10) +17684952456223*a(n-11) -257017179787974*a(n-12) +44697122178759*a(n-13) +6813950647173658*a(n-14) -14214676649780235*a(n-15) -95256883925367556*a(n-16) +349764223086150739*a(n-17) +638075361803414056*a(n-18) -4132669977915280075*a(n-19) -655758716192007656*a(n-20) +27891199596575003557*a(n-21) -19946115396154900446*a(n-22) -113398578310551386143*a(n-23) +153863761785049382215*a(n-24) +275337771685131610985*a(n-25) -574115913592410065960*a(n-26) -351972189331006256045*a(n-27) +1287605861871487083865*a(n-28) +49832950543007059662*a(n-29) -1835218083346443858712*a(n-30) +584632105931636070627*a(n-31) +1673748635702117796496*a(n-32) -1004898866351611047643*a(n-33) -943485141803302660797*a(n-34) +864920595743414931962*a(n-35) +287374596274394425107*a(n-36) -446718459459833923638*a(n-37) -15289668350134708829*a(n-38) +143310897427441778213*a(n-39) -21277189785541190392*a(n-40) -28130375346377645531*a(n-41) +8130899172562042318*a(n-42) +3152590800552865603*a(n-43) -1415161150801970578*a(n-44) -157323743807361158*a(n-45) +132817713592064303*a(n-46) -2466246093921598*a(n-47) -6544805823376510*a(n-48) +605341636474744*a(n-49) +143100794076816*a(n-50) -20407671803168*a(n-51) -733454285952*a(n-52) +152009496576*a(n-53)
EXAMPLE
Some solutions for n=3
..0..2..1..0..2..1....0..1..0..1..4..1....0..1..0..3..0..2....0..2..0..3..5..3
..0..2..1..0..2..1....0..3..4..1..4..1....0..1..0..2..0..2....0..2..0..2..5..3
..0..2..1..0..2..0....0..3..4..5..4..1....0..1..0..2..0..3....0..2..0..3..5..3
Petersen graph (3,1) coloring a rectangular array: number of nX7 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0
+10
1
729, 110427, 22598167, 4867038759, 1065016901935, 234215122981463, 51596648899152901, 11373354088592222347, 2507537188605388269479, 552889843504513864372699, 121910555703890549598868125
EXAMPLE
Some solutions for n=3
..0..1..0..1..2..5..4....0..1..0..1..0..3..5....0..1..0..1..0..3..4
..0..1..4..1..4..5..4....0..1..4..3..0..2..0....0..1..2..1..4..1..0
..0..1..2..1..4..1..4....0..1..0..1..0..3..0....0..1..4..1..2..1..4
Petersen graph (3,1) coloring a rectangular array: number of 2 X n 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.
+10
1
6, 19, 115, 631, 3539, 19759, 110427, 617015, 3447747, 19265087, 107648363, 601511175, 3361088979, 18780896143, 104942791931, 586393188311, 3276613524707, 18308869209055, 102305227390859, 571655159691687
FORMULA
Empirical: a(n) = 5*a(n-1) + 4*a(n-2) - 4*a(n-3) for n>4.
Empirical g.f.: x*(2 - x)*(1 - 2*x)*(3 + 2*x) / (1 - 5*x - 4*x^2 + 4*x^3). - Colin Barker, Aug 21 2018
EXAMPLE
Some solutions for n=3:
..0..1..0....0..3..0....0..2..0....0..2..1....0..2..1....0..1..4....0..1..0
..0..1..4....5..3..5....5..2..5....1..2..0....0..2..0....4..1..0....2..1..0
Petersen graph (3,1) coloring a rectangular array: number of 3 X n 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.
+10
1
36, 121, 1519, 16323, 182901, 2030665, 22598167, 251348043, 2795984857, 31101456601, 345963177427, 3848382739711, 42808185822221, 476184598157809, 5296925638013539, 58921311528252323, 655421879453116645
FORMULA
Empirical: a(n) = 12*a(n-1) - 4*a(n-2) - 73*a(n-3) + 103*a(n-4) - 23*a(n-5) - 16*a(n-6) + 4*a(n-7) for n>8.
Empirical g.f.: x*(36 - 311*x + 211*x^2 + 1207*x^3 - 1774*x^4 + 397*x^5 + 272*x^6 - 68*x^7) / (1 - 12*x + 4*x^2 + 73*x^3 - 103*x^4 + 23*x^5 + 16*x^6 - 4*x^7). - Colin Barker, Aug 21 2018
EXAMPLE
Some solutions for n=3:
..0..2..5....0..3..4....0..1..0....0..2..1....0..3..0....0..3..0....0..1..4
..1..2..1....0..3..5....4..1..4....0..2..1....5..2..0....4..1..0....4..3..4
..5..4..1....5..3..5....4..3..0....1..2..1....0..2..5....4..3..0....0..1..0
Petersen graph (3,1) coloring a rectangular array: number of 4Xn 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0
+10
1
216, 771, 20115, 426359, 9685063, 216562815, 4867038759, 109246101385, 2453094910375, 55078160026621, 1236680655855829, 27767207466078683, 623458974380912329, 13998557054872762899, 314310396038821269603
FORMULA
Empirical: a(n) = 25*a(n-1) +a(n-2) -1509*a(n-3) +3743*a(n-4) +21956*a(n-5) -87188*a(n-6) -23069*a(n-7) +409623*a(n-8) -235845*a(n-9) -749323*a(n-10) +679813*a(n-11) +599294*a(n-12) -680632*a(n-13) -199246*a(n-14) +294548*a(n-15) +14686*a(n-16) -53558*a(n-17) +3396*a(n-18) +3220*a(n-19) -192*a(n-20) -64*a(n-21) for n>22
EXAMPLE
Some solutions for n=3
..0..1..4....0..1..2....0..2..0....0..1..2....0..3..4....0..2..0....0..3..0
..4..1..2....4..1..0....0..2..1....0..1..2....5..3..5....0..2..0....0..3..4
..2..1..4....4..3..4....5..2..0....4..1..2....0..2..5....5..2..5....0..1..4
..4..1..2....0..1..0....1..2..5....4..5..2....5..3..5....0..2..1....2..1..4
Petersen graph (3,1) coloring a rectangular array: number of 5Xn 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0
+10
1
1296, 4913, 266419, 11148439, 515473927, 23328902821, 1065016901935, 48530437419865, 2213179954647275, 100913208621796747, 4601629002961862345, 209830596880154645775, 9568174653385280051091, 436303604544116583704607
FORMULA
Empirical: a(n) = 71*a(n-1) -1025*a(n-2) -14582*a(n-3) +432132*a(n-4) -1235038*a(n-5) -44254492*a(n-6) +375953458*a(n-7) +1077097488*a(n-8) -24108628735*a(n-9) +43813966193*a(n-10) +660782580981*a(n-11) -3015474264116*a(n-12) -7468946258468*a(n-13) +72313665742943*a(n-14) -19748204982172*a(n-15) -929976166077118*a(n-16) +1623691507031261*a(n-17) +6877758733216211*a(n-18) -21986547259066956*a(n-19) -25977258135841984*a(n-20) +164780020970184872*a(n-21) -5445523483934936*a(n-22) -789421436773000211*a(n-23) +617827785709579554*a(n-24) +2499061275173634960*a(n-25) -3608966242372275158*a(n-26) -5054385737333805739*a(n-27) +11913328661514326768*a(n-28) +5266973549905528132*a(n-29) -26083920461220425468*a(n-30) +2323512355364237888*a(n-31) +39474091164345616200*a(n-32) -18570564930623297944*a(n-33) -41144392421727062733*a(n-34) +34192277387530560380*a(n-35) +27957670701203653789*a(n-36) -37366923751687843813*a(n-37) -9816688145756259804*a(n-38) +27153007122450867062*a(n-39) -1474941418773154672*a(n-40) -13352938311422813034*a(n-41) +3795826701250077433*a(n-42) +4315925800274009339*a(n-43) -2185410858126306921*a(n-44) -825385995637082366*a(n-45) +704751836109259081*a(n-46) +54959242071750150*a(n-47) -139309166655413192*a(n-48) +12738975912252902*a(n-49) +16584925179127396*a(n-50) -3592712328375964*a(n-51) -1071918437017524*a(n-52) +385473088083924*a(n-53) +24628625488560*a(n-54) -20197202519736*a(n-55) +763271541072*a(n-56) +473574597408*a(n-57) -42448453056*a(n-58) -3476245248*a(n-59) +407586816*a(n-60) for n>61
EXAMPLE
Some solutions for n=3
..0..2..0....0..2..1....0..1..0....0..1..2....0..2..1....0..1..0....0..1..0
..1..2..1....1..2..5....0..2..0....4..1..0....0..2..0....2..1..4....0..3..4
..1..2..1....0..2..1....1..2..5....2..1..2....0..2..1....2..1..2....4..3..5
..1..2..0....5..2..5....5..2..5....4..5..4....0..2..0....2..5..4....0..3..0
..5..2..1....5..2..5....5..3..5....3..5..4....5..2..0....4..5..4....0..3..4
Petersen graph (3,1) coloring a rectangular array: number of 6Xn 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0
+10
1
7776, 31307, 3528715, 291545903, 27465794119, 2519813048575, 234215122981463, 21722081604000233, 2017473470471496373, 187345647840479535879, 17400111813793245517801, 1616059549379820468437485
EXAMPLE
Some solutions for n=3
..0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0
..0..2..0....0..1..2....0..2..0....0..1..0....0..1..2....0..1..0....0..1..4
..0..2..0....2..1..2....1..2..1....2..1..0....0..1..4....2..1..2....2..1..2
..5..3..0....4..5..2....1..2..0....4..1..0....4..1..4....2..1..4....4..5..4
..0..2..5....4..5..2....0..2..1....4..3..4....4..1..4....2..1..2....2..1..2
..1..2..0....4..1..4....1..2..1....4..1..4....0..1..4....4..1..0....2..5..4
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