A384634
Expansion of (1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4).
Original entry on oeis.org
1, 2, 7, 16, 48, 120, 338, 880, 2412, 6392, 17316, 46240, 124640, 333920, 898168, 2409600, 6475408, 17382432, 46694512, 125377024, 336745984, 904275328, 2428594976, 6521881856, 17515179200, 47037120384, 126321412672, 339239675392, 911046599168, 2446649462272
Offset: 0
a(2)=7 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-1-4, 2-3-1, 2-3-2, 2-3-4.
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a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <0|1|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4), {x, 0, 32}], x]
A384635
Expansion of (1+3*x+2*x^2) / (1-6*x^2-4*x^3+2*x^4).
Original entry on oeis.org
1, 3, 8, 22, 58, 158, 420, 1136, 3036, 8180, 21920, 58952, 158168, 425032, 1140976, 3064960, 8229648, 22103600, 59355776, 159410272, 428089760, 1149677536, 3087468096, 8291603712, 22267339200, 59800139584, 160595513856, 431286986880, 1158238963072
Offset: 0
a(2)=8 because we have the walks 3-1-0, 3-1-2, 3-1-3. 3-1-4, 3-2-1, 3-2-3, 3-4-1, 3-4-3.
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a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <0|1|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4), {x, 0, 32}], x]
A384678
Expansion of (1+x) / (1-2*x-4*x^2+2*x^3).
Original entry on oeis.org
1, 3, 10, 30, 94, 288, 892, 2748, 8488, 26184, 80824, 249408, 769744, 2375472, 7331104, 22624608, 69822688, 215481600, 665004736, 2052290496, 6333636736, 19546425984, 60322817920, 186164066304, 574526552320, 1773063734016, 5471905544704, 16887012920832
Offset: 0
a(2)=10 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-1-4, 2-3-1, 2-3-2, 2-3-4, 2-4-1, 2-4-2, 2-4-3.
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a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1+x) / (1-2*x-4*x^2+2*x^3), {x, 0, 32}], x]
LinearRecurrence[{2,4,-2},{1,3,10},30] (* Harvey P. Dale, Jul 07 2025 *)
A384677
Expansion of (1-x-2*x^2) / (1-2*x-4*x^2+2*x^3).
Original entry on oeis.org
1, 1, 4, 10, 34, 100, 316, 964, 2992, 9208, 28456, 87760, 270928, 835984, 2580160, 7962400, 24573472, 75836224, 234041536, 722281024, 2229055744, 6879152512, 21229965952, 65518430464, 202198419712, 624010629376, 1925778076672, 5943201831424, 18341494710784
Offset: 0
a(3)=10 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-2-4, 0-1-3-1, 0-1-3-2, 0-1-3-4, 0-1-4-1, 0-1-4-2, 0-1-4-3.
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a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1-x-2*x^2) / (1-2*x-4*x^2+2*x^3), {x, 0, 32}], x]
Table[(MatrixPower[{{0,1,0,0,0},{1,0,1,1,1},{0,1,0,1,1},{0,1,1,0,1},{0,1,1,1,0}},n].{1,1,1,1,1}),{n,0,28}][[All,1]] (* Shenghui Yang, Jun 07 2025 *)
Showing 1-4 of 4 results.
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