Jeroen F.J. Laros has authored 5 sequences.
A101169
Lengths of successive words (starting with a) under the substitution: {a -> aab, b -> aac, c -> ab}.
Original entry on oeis.org
1, 3, 9, 26, 76, 221, 644, 1875, 5461, 15903, 46314, 134876, 392791, 1143896, 3331289, 9701475, 28252921, 82278978, 239615244, 697814501, 2032195756, 5918219771, 17235212309, 50192888175, 146173193506, 425689839228
Offset: 0
a => aab => aabaabaac => aabaabaacaabaabaacaabaabab, thus a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 26.
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a:= n-> (<<0|1|0>, <0|0|1>, <-1|3|2>>^n. <<0, 1, 3>>)[2,1]:
seq(a(n), n=0..30); # Alois P. Heinz, May 06 2011
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LinearRecurrence[{2,3,-1},{1,3,9},40] (* Harvey P. Dale, Feb 19 2012 *)
A101168
Lengths of successive words (starting with a) under the substitution: {a -> aab, b -> aac, c -> a}.
Original entry on oeis.org
1, 3, 9, 25, 71, 201, 569, 1611, 4561, 12913, 36559, 103505, 293041, 829651, 2348889, 6650121, 18827671, 53304473, 150914409, 427265435, 1209664161, 3424773601, 9696140959, 27451493281, 77720042081, 220039211683, 622970000809, 1763738467065, 4993456147431
Offset: 0
a => aab => aabaabaac => aabaabaacaabaabaacaabaaba, thus a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25.
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a:= n-> (<<0|1|0>, <0|0|1>, <1|2|2>>^n. <<1, 3, 9>>)[1, 1]:
seq(a(n), n=0..30); # Alois P. Heinz, May 06 2011
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Length/@SubstitutionSystem[{a->{a,a,b},b->{a,a,c},c->a},{a},15] (* The program generates the first 16 terms of the sequence. To generate more, increase the final ("15") constant. *) (* Harvey P. Dale, Sep 05 2022 *)
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a(n):=b(n+1);
b(n):= sum(sum((sum(binomial(j,n+1-m-3*k+2*j) *binomial(k,j), j,0, k)) *sum(binomial(i,m-i) *binomial(k+i-1,k-1),i,ceiling(m/2),m), m,0, n+1-k), k,1,n+1); /* Vladimir Kruchinin, May 05 2011 */
A101197
Lengths of successive words (starting with a) under the substitution: {a -> aab, b -> aac, c -> b}.
Original entry on oeis.org
1, 3, 9, 25, 71, 199, 561, 1577, 4439, 12487, 35137, 98857, 278151, 782599, 2201937, 6195369, 17431351, 49044935, 137993185, 388258473, 1092406631, 3073602311, 8647907569, 24331808809, 68460135703, 192619882695
Offset: 0
a => aab => aabaabaac => aabaabaacaabaabaacaabaabb, thus a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25.
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a:= n-> (<<0|1|0>, <0|0|1>, <-2|3|2>>^n. <<0, 1, 3>>)[2, 1]:
seq (a(n), n=0..30); # Alois P. Heinz, May 06 2011
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LinearRecurrence[{2,3,-2},{1,3,9},30] (* Harvey P. Dale, Jan 01 2019 *)
A101400
a(n) = a(n-1) + 2*a(n-2) + a(n-3) - a(n-4).
Original entry on oeis.org
1, 2, 5, 10, 21, 44, 91, 190, 395, 822, 1711, 3560, 7409, 15418, 32085, 66770, 138949, 289156, 601739, 1252230, 2605915, 5422958, 11285279, 23484880, 48872481, 101704562, 211649125, 440445850, 916576181, 1907412444, 3969361531
Offset: 0
a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 10, a(4) = 21, a(5) = 44
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a:=[1,2,5,10];; for n in [5..35] do a[n]:=a[n-1]+2*a[n-2]+a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Apr 03 2018
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I:=[1,2,5,10]; [n le 4 select I[n] else Self(n-1) + 2*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Apr 03 2018
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m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x+x^2)/(1-x-2*x^2-x^3+x^4))); // G. C. Greubel, Apr 03 2018
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a[0] = 1; a[1] = 2; a[2] = 5; a[3] = 10; a[n_] := a[n] = a[n - 1] + 2a[n - 2] + a[n - 3] - a[n - 4]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 15 2005 *)
LinearRecurrence[{1,2,1,-1},{1,2,5,10},40] (* Harvey P. Dale, Oct 24 2017 *)
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x='x+O('x^30); Vec((1+x+x^2)/(1-x-2*x^2-x^3+x^4)) \\ G. C. Greubel, Apr 03 2018
A101399
a(0) = 1, a(1) = 2, a(2) = 5; for n >= 3, a(n) = a(n-1) + 2*a(n-2) + a(n-3).
Original entry on oeis.org
1, 2, 5, 10, 22, 47, 101, 217, 466, 1001, 2150, 4618, 9919, 21305, 45761, 98290, 211117, 453458, 973982, 2092015, 4493437, 9651449, 20730338, 44526673, 95638798, 205422482, 441226751, 947710513, 2035586497, 4372234274
Offset: 0
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a:=[1,2,5];; for n in [4..35] do a[n]:=a[n-1]+2*a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Apr 03 2018
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I:=[1,2,5]; [n le 3 select I[n] else Self(n-1) + 2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 03 2018
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m:=25; R:=PowerSeriesRing( Integers(), m); Coefficients(R!((1+x+x^2)/(1-x-2*x^2-x^3))); // G. C. Greubel, Apr 03 2018
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a[0] = 1; a[1] = 2; a[2] = 5; a[n_] := a[n] = a[n - 1] + 2a[n - 2] + a[n - 3]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 15 2005 *)
LinearRecurrence[{1,2,1},{1,2,5},30] (* Harvey P. Dale, Aug 29 2012 *)
CoefficientList[Series[(1+x+x^2)/(1-x-2*x^2-x^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 03 2018 *)
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x='x+O('x^30); Vec((1+x+x^2)/(1-x-2*x^2-x^3)) \\ G. C. Greubel, Apr 03 2018
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