A384672
Expansion of (1+2*x-x^2) / (1-2*x-5*x^2+2*x^3).
Original entry on oeis.org
1, 4, 12, 42, 136, 458, 1512, 5042, 16728, 55642, 184840, 614434, 2041784, 6786058, 22552168, 74951058, 249090840, 827832634, 2751217352, 9143416194, 30387253880, 100989154026, 335627745064, 1115426752498, 3707013922264, 12319906116890, 40944028340104
Offset: 0
a(2)=12 because we have the walks 1-0-1, 1-0-4, 1-2-1, 1-2-3, 1-2-4, 1-3-1, 1-3-2, 1-3-4, 1-4-0, 1-4-1, 1-4-2, 1-4-3.
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a:= n-> (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <1|1|1|1|0>>^n. <<1,1,1,1,1>>)[2,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1+2*x-x^2) / (1-2*x-5*x^2+2*x^3), {x, 0, 32}], x]
LinearRecurrence[{2,5,-2},{1,4,12},30] (* Harvey P. Dale, Aug 30 2025 *)
A384647
Expansion of (1+3*x+x^2) / (1-x-5*x^2-2*x^3).
Original entry on oeis.org
1, 4, 10, 32, 90, 270, 784, 2314, 6774, 19912, 58410, 171518, 503392, 1477802, 4337798, 12733592, 37378186, 109721742, 322079856, 945444938, 2775287702, 8146672104, 23914000490, 70197936414, 206061283072, 604878966122, 1775581254310, 5212098651064
Offset: 0
a(2)=10 because we have the walks 1-0-1, 1-0-4, 1-2-1, 1-2-3, 1-3-1, 1-3-2, 1-3-4, 1-4-0, 1-4-1, 1-4-3.
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a:= n-> (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <1|1|0|1|0>>^n. <<1,1,1,1,1>>)[2,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1+3*x+x^2) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]
A384648
Expansion of (1+2*x+x^2) / (1-x-5*x^2-2*x^3).
Original entry on oeis.org
1, 3, 9, 26, 77, 225, 662, 1941, 5701, 16730, 49117, 144169, 423214, 1242293, 3646701, 10704594, 31422685, 92239057, 270761670, 794802325, 2333088789, 6848623754, 20103672349, 59012968697, 173228577950, 508500766133, 1492669593277, 4381630579842
Offset: 0
a(2)=9 because we have the walks 3-1-0, 3-1-2, 3-1-3, 3-1-4, 3-2-1, 3-2-3, 3-4-0, 3-4-1, 3-4-3.
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a:= n-> (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <1|1|0|1|0>>^n. <<1,1,1,1,1>>)[4,1]:
seq(a(n), n=0..32);
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CoefficientList[Series[(1+2*x+x^2) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]
A384671
Expansion of (1-x^2) / (1-2*x-5*x^2+2*x^3).
Original entry on oeis.org
1, 2, 8, 24, 84, 272, 916, 3024, 10084, 33456, 111284, 369680, 1228868, 4083568, 13572116, 45104336, 149902116, 498181680, 1655665268, 5502434704, 18286832388, 60774507760, 201978308052, 671255490128, 2230853504996, 7414027844528, 24639812233780
Offset: 0
a(2)=8 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-4-0, 0-4-1, 0-4-2, 0-4-3.
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a:= n-> (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <1|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) / (1-2*x-5*x^2+2*x^3), {x, 0, 32}], x]
A384673
Expansion of (1+x) / (1-2*x-5*x^2+2*x^3).
Original entry on oeis.org
1, 3, 11, 35, 119, 391, 1307, 4331, 14415, 47871, 159155, 528835, 1757703, 5841271, 19413387, 64517723, 214419839, 712601519, 2368266787, 7870701491, 26157533879, 86932041639, 288910349691, 960165839819, 3191019344815, 10605047189343, 35244859423123
Offset: 0
a(2)=11 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, 2-4-0, 2-4-1, 2-4-2, 2-4-3.
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a:= n-> (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <1|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+x) / (1-2*x-5*x^2+2*x^3), {x, 0, 32}], x]
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