A343529 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, X, Y.
1, 0, 2, 4, 18, 36, 138, 334, 1066, 3096, 9490, 26826, 80468, 235718, 699056, 2055466, 6074498, 17857906, 52725190, 155445504, 458505084, 1351257730, 3984941402, 11748306100, 34643781158, 102144907886, 301179533022, 887996181502, 2618324249106, 7720149428450
Offset: 0
Examples
a(2) = 2, a(3) = 4: a(5) = 36: .___. .___. ._____. ._________. ._________. | | | | |_. | | |_. | |_. ._._| | | ._| |_. | | |___| | ._| |___| | |_| |_. | |_| | | |_| | ._| | |_|_. ._| | | |_. ._| | | | | | | | | (4) | |_| | (2) | ._|_| |_| (2) ... |___| |___| |_|___| |_____|___| |_|_______| . . a(4) = 18: .___.___. .___.___. ._._____. ._______. | | | | | | | |_. | |___. ._| | ._|_. | | ._| ._| | |___| | |_| | |_| | |_| |_| |_| | |___| | | ._| | | | | (2) | | | (2) | |_. | (2) |_| |___| (2) |___|___| |___|___| |_____|_| |_______| . ._______. ._._____. ._______. | | ._| | |_. | |___. ._| | |___| | | ._|___| | |_| | | ._|_. | | |_. | | ._|_. | |_| | | (2) |_| |___| (4) |_| | | (4) |_____|_| |_______| |_____|_| .
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..2131
- Wikipedia, Pentomino
- Index entries for linear recurrences with constant coefficients, signature (0, 4, 7, 16, 21, 19, -61, -198, -153, 399, 172, -511, -223, -1843, -1515, -702, 1445, 880, -6060, -13769, -3634, -761, -2886, 6196, -11350, -1559, -24546, 5474, 54995, 15927, -30841, 9893, 7921, 68397, -41768, -48989, 36892, 61021, 7095, -47120, -56180, 24792, 34252, 8426, -13154, -11260, 4136, 5668, -2228, -848, -1024, 224, 224, 144).
Formula
G.f.: (16*x^54 +32*x^53 -128*x^51 -80*x^50 +380*x^49 +540*x^48 +456*x^47 -1316*x^46 -28*x^45 +976*x^44 +6016*x^43 +3356*x^42 -1680*x^41 -5992*x^40 -919*x^39 -825*x^38 +5838*x^37 -12209*x^36 -14876*x^35 -17029*x^34 -15243*x^33 -13879*x^32 -8029*x^31 -17115*x^30 -3713*x^29 -6022*x^28 -110*x^27 +1321*x^26 -832*x^25 -212*x^24 +4478*x^23 -575*x^22 -808*x^21 -3929*x^20 -574*x^19 +314*x^18 -1001*x^17 -1354*x^16 -805*x^15 -493*x^14 -299*x^13 -229*x^12 -78*x^11 +177*x^10 -39*x^9 -50*x^8 -19*x^7 +13*x^6 +15*x^5 +6*x^4 +3*x^3 +2*x^2 -1) /
(144*x^54 +224*x^53 +224*x^52 -1024*x^51 -848*x^50 -2228*x^49 +5668*x^48 +4136*x^47 -11260*x^46 -13154*x^45 +8426*x^44 +34252*x^43 +24792*x^42 -56180*x^41 -47120*x^40 +7095*x^39 +61021*x^38 +36892*x^37 -48989*x^36 -41768*x^35 +68397*x^34 +7921*x^33 +9893*x^32 -30841*x^31 +15927*x^30 +54995*x^29 +5474*x^28 -24546*x^27 -1559*x^26 -11350*x^25 +6196*x^24 -2886*x^23 -761*x^22 -3634*x^21 -13769*x^20 -6060*x^19 +880*x^18 +1445*x^17 -702*x^16 -1515*x^15 -1843*x^14 -223*x^13 -511*x^12 +172*x^11 +399*x^10 -153*x^9 -198*x^8 -61*x^7 +19*x^6 +21*x^5 +16*x^4 +7*x^3 +4*x^2 -1).