A255256
Number A(n,k) of 2-colorings of a k X n rectangle such that no nontrivial subsquare has monochromatic corners; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 50, 50, 16, 1, 1, 32, 178, 276, 178, 32, 1, 1, 64, 634, 1498, 1498, 634, 64, 1, 1, 128, 2258, 8352, 10980, 8352, 2258, 128, 1, 1, 256, 8042, 46730, 85138, 85138, 46730, 8042, 256, 1, 1, 512, 28642, 260204, 655090, 781712, 655090, 260204, 28642, 512, 1
Offset: 0
A(2,2) = 2^(2*2) - 2 = 14 because there are exactly two of sixteen 2-colorings of the 2 X 2 square resulting in nontrivial subsquares with monochromatic corners.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 4, 8, 16, 32, 64, ...
1, 4, 14, 50, 178, 634, 2258, ...
1, 8, 50, 276, 1498, 8352, 46730, ...
1, 16, 178, 1498, 10980, 85138, 655090, ...
1, 32, 634, 8352, 85138, 781712, 6965108, ...
1, 64, 2258, 46730, 655090, 6965108, 58339148, ...
A133129
Number of black/white colorings of a 3 X n rectangle which have no monochromatic 2 by 2 subsquares.
Original entry on oeis.org
1, 8, 50, 322, 2066, 13262, 85126, 546410, 3507314, 22512862, 144506294, 927561722, 5953863490, 38216853518, 245307588134, 1574588362378, 10107019231634, 64875265300670, 416423472774166, 2672952594083738, 17157235452223586, 110129423550044398
Offset: 0
a(2) = 50 because if the middle row is not monochromatic, the top and bottom rows are unconstrained, contributing 2*4*4. if the middle row is monochromatic, the top and bottom rows can each take on only 3 values contributing 2*3*3.
A255255
Number of 2-colorings of a 4 X n rectangle such that no nontrivial subsquare has monochromatic corners.
Original entry on oeis.org
1, 16, 178, 1498, 10980, 85138, 655090, 5115398, 39914386, 312388874, 2436283602, 18994966598, 148059349634, 1154792660474, 9007078544234, 70254124462638, 547921292697778, 4273303250042966, 33327954035543034, 259932116476519958, 2027268764564330754
Offset: 0
-
gf:= -(4608*x^78 +3840*x^77 +84224*x^76 -320640*x^75 +246976*x^74 -756544*x^73 -2238400*x^72 +5321120*x^71 +12808672*x^70 +54862128*x^69 -56935120*x^68 -320396192*x^67 +196696056*x^66 +580449400*x^65 -800166640*x^64 +931676252*x^63 -3217094764*x^62 -2931282696*x^61 +15075340414*x^60 -25189807228*x^59 +8940907182*x^58 +72225838335*x^57 -79785446783*x^56 +16049408070*x^55 +94923048557*x^54 -184157650742*x^53 +84396466384*x^52 +63222211250*x^51 -205358037404*x^50
+102077559913*x^49 -54179691207*x^48 -58933050614*x^47 -72350094400*x^46 +119755954460*x^45 +391109674279*x^44 -31136120454*x^43 +7469466171*x^42 +4087734421*x^41 -259850982087*x^40 -250072129598*x^39 -106581496436*x^38 +208831935210*x^37 +402861489180*x^36 +109203162981*x^35 -275403863093*x^34 -334505242945*x^33 -114680900580*x^32 +196805363764*x^31 +294322652328*x^30 +163414286266*x^29
+9671161820*x^28 -93783726011*x^27 -117305441726*x^26 -63470276869*x^25 -4776176481*x^24 +17768047304*x^23 +14651146288*x^22 +7623498972*x^21 +4260618627*x^20 +1317986665*x^19 -720102442*x^18 -787824686*x^17 -195158015*x^16 +30687326*x^15 +15478943*x^14 +9512482*x^13 +14257207*x^12 +7365310*x^11 -194075*x^10 -1591059*x^9 -609139*x^8 -99289*x^7 +358*x^6 +4060*x^5 +1728*x^4 +632*x^3 +117*x^2 +14*x +1) /
(-2304*x^76 -4224*x^75 -48832*x^74 +106368*x^73 -28736*x^72 +387264*x^71 +1606752*x^70 -417728*x^69 -6732224*x^68 -32605960*x^67 -17678016*x^66 +104280356*x^65 +13666240*x^64 -287868964*x^63 +243731588*x^62 -345759930*x^61 +338343404*x^60 +2718472224*x^59 -4711903718*x^58 +5420902529*x^57 +8555023111*x^56 -23454331276*x^55 +9026831269*x^54
+16445485090*x^53 -42633850200*x^52 +27124456832*x^51 +15867136764*x^50 -59350410523*x^49 +30378083319*x^48 +11917445228*x^47 -36411691280*x^46 +24780724314*x^45 +72006964843*x^44 +20800791816*x^43 -64959622007*x^42 -16674723587*x^41 +8311317841*x^40 -41851983658*x^39 -10878840512*x^38 +60227329766*x^37 +71646302640*x^36 +111795917*x^35 -98969183331*x^34 -102248503015*x^33 +13126172584*x^32 +108338140330*x^31
+80553248220*x^30 -4243013616*x^29 -43486302952*x^28 -28181948095*x^27 -3688188878*x^26 +4299265271*x^25 +2263896509*x^24 +739802084*x^23 +845394966*x^22 +509931484*x^21 -127596363*x^20 -330548479*x^19 -74104054*x^18 +132481026*x^17 +79485565*x^16 +21861608*x^15 +12380793*x^14 +5359440*x^13 +161495*x^12 -1832730*x^11 -1066905*x^10 -274895*x^9 -73293*x^8 -3043*x^7 +5244*x^6 +1760*x^5 +358*x^4 +46*x^3 +29*x^2 +2*x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);
Showing 1-3 of 3 results.