A384641
Expansion of (1+2*x-x^3) / (1-x-5*x^2+x^3+2*x^4).
Original entry on oeis.org
1, 3, 8, 21, 56, 147, 390, 1027, 2718, 7169, 18952, 50025, 132180, 349015, 921986, 2434831, 6431386, 16985525, 44863652, 118490229, 312960192, 826576635, 2183160062, 5766102587, 15229405878, 40223605481, 106238212160, 280594628513, 741103272076, 1957390991519
Offset: 0
a(2)=8 because we have the walks 2-1-0, 2-1-2, 2-1-4, 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|0|1>, <0|1|0|1|1>, <0|0|1|0|1>, <0|1|1|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^3) / (1-x-5*x^2+x^3+2*x^4), {x, 0, 32}], x]
A384642
Expansion of (1+x-x^2+x^3) / (1-x-5*x^2+x^3+2*x^4).
Original entry on oeis.org
1, 2, 6, 16, 42, 112, 294, 780, 2054, 5436, 14338, 37904, 100050, 264360, 698030, 1843972, 4869662, 12862772, 33971050, 89727304, 236980458, 625920384, 1653153270, 4366320124, 11532205174, 30458811756, 80447210962, 212476424320, 561189257026, 1482206544152
Offset: 0
a(2)=6 because we have the walks 3-2-1, 3-2-3, 3-2-4, 3-4-1, 3-4-2, 3-4-3.
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a:= n-> (<<0|1|0|0|0>, <1|0|1|0|1>, <0|1|0|1|1>, <0|0|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[4,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1+x-x^2+x^3) / (1-x-5*x^2+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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