A384598
Expansion of (1-3*x^2) / (1-x-4*x^2+2*x^3+2*x^4).
Original entry on oeis.org
1, 1, 2, 4, 8, 18, 38, 86, 186, 418, 914, 2042, 4490, 9994, 22042, 48954, 108154, 239898, 530522, 1175898, 2601882, 5764634, 12759322, 28262298, 62566554, 138567834, 306790810, 679404442, 1504298906, 3331199386, 7376004506, 16333395354, 36166416794
Offset: 0
a(3)=4 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-2-4.
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a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-2|-2|4|1>>^n. <<1,1,2,4>>)[1,1]:
seq(a(n), n=0..32); # Alois P. Heinz, Jun 04 2025
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CoefficientList[Series[(1 - 3*x^2)/(1 - x - 4*x^2 + 2*x^3 + 2*x^4), {x, 0, 32}], x] (* Michael De Vlieger, Jun 04 2025 *)
A384599
Expansion of (1+3*x+2*x^2) / (1-4*x^2-2*x^3).
Original entry on oeis.org
1, 3, 6, 14, 30, 68, 148, 332, 728, 1624, 3576, 7952, 17552, 38960, 86112, 190944, 422368, 936000, 2071360, 4588736, 10157440, 22497664, 49807232, 110305536, 244224256, 540836608, 1197508096, 2651794944, 5871705600, 13002195968, 28790412288, 63752195072
Offset: 0
a(2)=6 because we have the walks 2-1-0, 2-1-2, 2-3-2, 2-3-4, 2-4-2, 2-4-3.
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a:= n-> (<<0|1|0>, <0|0|1>, <2|4|0>>^n. <<1,3,6>>)[1,1]:
seq(a(n), n=0..31); # Alois P. Heinz, Jun 04 2025
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CoefficientList[Series[(1 + 3*x + 2*x^2)/(1 - 4*x^2 - 2*x^3), {x, 0, 31}], x] (* Michael De Vlieger, Jun 04 2025 *)
A384600
Expansion of (1+x-x^2) / (1-x-4*x^2+2*x^3+2*x^4).
Original entry on oeis.org
1, 2, 5, 11, 25, 55, 123, 271, 603, 1331, 2955, 6531, 14483, 32035, 70995, 157107, 348051, 770419, 1706419, 3777779, 8366515, 18523955, 41021619, 90828851, 201134387, 445358643, 986195251, 2183703347, 4835498291, 10707203891, 23709399859, 52499812147
Offset: 0
a(2)=5 because we have the walk 3-2-1, 3-2-3, 3-2-4, 3-4-2, 3-4-3.
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a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-2|-2|4|1>>^n. <<1,2,5,11>>)[1,1]:
seq(a(n), n=0..31); # Alois P. Heinz, Jun 04 2025
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CoefficientList[Series[(1 + x - x^2)/(1 - x - 4*x^2 + 2*x^3 + 2*x^4), {x, 0, 31}], x] (* Michael De Vlieger, Jun 04 2025 *)
A384604
Expansion of (1-x^2) / (1-x-4*x^2+2*x^3).
Original entry on oeis.org
1, 1, 4, 6, 20, 36, 104, 208, 552, 1176, 2968, 6568, 16088, 36424, 87640, 201160, 478872, 1108232, 2621400, 6096584, 14365720, 33509256, 78778968, 184084552, 432181912, 1010962184, 2371520728, 5551005640, 13015164184, 30476145288, 71434790744, 167309043528
Offset: 0
a(3)=6 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-3-1, 0-1-3-2, 0-1-4-1.
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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|0>, <0|1|0|0|0>>^n. <<1,1,1,1,1>>)[1,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1 - x^2)/(1 - x - 4*x^2 + 2*x^3), {x, 0, 32}], x]
A384605
Expansion of (1+x) / (1-x-4*x^2+2*x^3).
Original entry on oeis.org
1, 2, 6, 12, 32, 68, 172, 380, 932, 2108, 5076, 11644, 27732, 64156, 151796, 352956, 831828, 1940060, 4561460, 10658044, 25023764, 58533020, 137311988, 321396540, 753578452, 1764540636, 4136061364, 9687067004, 22702231188, 53178376476, 124613167220
Offset: 0
a(3)=6 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-3-1, 0-1-3-2, 0-1-4-1.
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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|0>, <0|1|0|0|0>>^n. <<1,1,1,1,1>>)[3,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1 + x)/(1 - x - 4*x^2 + 2*x^3), {x, 0, 32}], x]
A384614
Expansion of (1+x+x^2) / (1-x-3*x^2).
Original entry on oeis.org
1, 2, 6, 12, 30, 66, 156, 354, 822, 1884, 4350, 10002, 23052, 53058, 122214, 281388, 648030, 1492194, 3436284, 7912866, 18221718, 41960316, 96625470, 222506418, 512382828, 1179902082, 2717050566, 6256756812, 14407908510, 33178178946, 76401904476, 175936441314
Offset: 0
a(2)=6 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-3-1, 2-3-2, 2-3-4.
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a:= n-> (<<0|1|0|0|0>, <1|0|1|1|0>, <0|1|0|1|0>, <0|1|1|0|1>, <0|0|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1+x+x^2) / (1-x-3*x^2), {x, 0, 32}], x]
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