A135364 -id:A135364 - cp's OEIS frontend

A097550 Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.

1, 3, 8, 19, 44, 102, 237, 551, 1281, 2978, 6923, 16094, 37414, 86977, 202197, 470051, 1092736, 2540303, 5905488, 13728594, 31915109, 74193627, 172479257, 400965626, 932131991, 2166943978, 5037533578, 11710844769, 27224411129, 63289077427

Offset: 0

D n Verma, Aug 16 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Cf. A097551, A097552, A097553, A097554, A097555, A097556.

Cf. A095263, A135364, A136302, A136303, A136304, A136305, A137229, A137234, A137249.

[n le 3 select Fibonacci(2*n) else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Apr 19 2021

a:= n-> (<<1|1|2>>. <<3|1|0>, <-2|0|1>, <1|0|0>>^n)[1$2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 24 2008

LinearRecurrence[{3,-2,1},{1,3,8},30] (* Harvey P. Dale, Jul 10 2019 *)

@CachedFunction
def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) )
def A097550(n): return A095263(n) +A095263(n-2)
[A097550(n) for n in (0..30)] # G. C. Greubel, Apr 19 2021

G.f.: (1+x^2)/(1 - 3*x+ 2*x^2 - x^3).
a(n) = term (1,1) in the 1 X 3 matrix [1,1,2].[3,1,0; -2,0,1; 1,0,0]^n. - Alois P. Heinz, Jul 24 2008
a(n) = A095263(n) + A095263(n-2). - G. C. Greubel, Apr 19 2021

More terms from Ryan Propper, Sep 27 2005

A136303 Expansion of g.f. (1 +x^2)/((1-x)^2(1 -3x +2*x^2 -x^3)).

Original entry on oeis.org

1, 5, 17, 48, 123, 300, 714, 1679, 3925, 9149, 21296, 49537, 115192, 267824, 622653, 1447533, 3365149, 7823068, 18186475, 42278476, 98285586, 228486323, 531166317, 1234811937, 2870589548, 6673311137, 15513566304, 36064666240, 83840177305

Offset: 0

Richard Choulet, Mar 22 2008

Previous name: Transform of 0 by the reciprocal transformation to T_{1,1} (see link).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (5,-9,8,-4,1).

Cf. A095263, A097550, A135364, A136302, A136304, A136305, A137229, A137234, A137249.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x^2)/((1-x)^2*(1-3*x+2*x^2-x^3)) )); // G. C. Greubel, Apr 19 2021

A136303:= n-> -2*(n+2) + add( (5*binomial(n+k+2, 3*k+2) - 4*binomial(n +k+1, 3*k+2) + 2*binomial(n+k, 3*k+2)), k=0..n/2 );
seq(A136303(n), n=0..40); # G. C. Greubel, Apr 19 2021

LinearRecurrence[{5,-9,8,-4,1},{1,5,17,48,123},40] (* Harvey P. Dale, Apr 01 2018 *)

def A136303_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)/((1-x)^2*(1-3*x+2*x^2-x^3)) ).list()
A136303_list(40) # G. C. Greubel, Apr 19 2021

G.f.: f(z) = 1 +5*z + ... = (1+z^2)/((1-z)^2*(1-3*z+2*z^2-z^3)).
a(n+5) = 5*a(n+4) -9*a(n+3) +8*a(n+2) -4*a(n+1) +a(n) (n>=0). - Richard Choulet, Apr 07 2009
From G. C. Greubel, Apr 19 2021: (Start)
a(n) = -2*(n+2) + 5*A095263(n) - 4*A095263(n-1) + 2*A095263(n-2).
a(n) = -2*(n+2) + Sum_{k=0..floor(n/2)} (5*binomial(n+k+2, 3*k+2) - 4*binomial(n +k+1, 3*k+2) + 2*binomial(n+k, 3*k+2)). (End)

A136304 Expansion of g.f. (1-z)^2(1-sqrt(1-4z))/(2z(1 - 3z + 2z^2 - z^3)).

Original entry on oeis.org

1, 2, 5, 14, 40, 116, 344, 1047, 3273, 10500, 34503, 115838, 396244, 1377221, 4851665, 17285662, 62173297, 225424527, 822919439, 3021713140, 11151957809, 41340655956, 153853915410, 574593145517, 2152679745351, 8087904580883, 30466311814036, 115036597198845

Offset: 0

Richard Choulet, Mar 22 2008

Previous name was: Transform of A000108 by the T_{0,0} transformation (see link).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard Choulet, Curtz-like transformation.

Cf. A097550, A135364, A136302, A136303, A136305, A137229, A137234, A137249.

Cf. A000108, A034943.

A034943:= func< n | (&+[Binomial(n+j-1, 3*j): j in [0..Floor(n/2)]]) >;
[(&+[A034943(j+1)*Catalan(n-j): j in [0..n]]): n in [0..35]]; // G. C. Greubel, Apr 19 2021

A034943[n_]:= A034943[n]= Sum[Binomial[n+k-1, 3*k], {k, 0, n/2}];
Table[Sum[A034943[j+1]*CatalanNumber[n-j], {j,0,n}], {n,0,35}] (* G. C. Greubel, Apr 19 2021 *)

def A034943(n): return sum(binomial(n+j-1,3*j) for j in (0..n//2))
[sum(A034943(j+1)*catalan_number(n-j) for j in (0..n)) for n in (0..35)] # G. C. Greubel, Apr 19 2021

G.f.: (1-z)^2*(1-sqrt(1-4*z))/(2*z*(1 - 3*z + 2*z^2 - z^3)).
Conjecture: (n+1)*a(n) + (-8*n+1)*a(n-1) + 3*(7*n-8)*a(n-2) + (-23*n+49)*a(n-3) + (13*n-32)*a(n-4) + 2*(-2*n+7)*a(n-5) = 0. - R. J. Mathar, Feb 29 2016
a(n) = Sum_{j=0..n} A034943(j+1)*A000108(n-j). - G. C. Greubel, Apr 19 2021

New name using g.f., Joerg Arndt, Apr 20 2021

A136305 Expansion of g.f. (3 -x +2x^2)/(1 -3x +2*x^2 -x^3).

Original entry on oeis.org

3, 8, 20, 47, 109, 253, 588, 1367, 3178, 7388, 17175, 39927, 92819, 215778, 501623, 1166132, 2710928, 6302143, 14650705, 34058757, 79177004, 184064203, 427897358, 994740672, 2312491503, 5375890523, 12497429235, 29052998162, 67540026539, 157011512528

Offset: 0

Richard Choulet, Mar 22 2008

Previous name: Transform of A000027 by the T_{1,2} transformation (see link).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Cf. A097550, A135364, A136302, A136303, A136304, A137229, A137234, A137249.

[n le 3 select 2^(n-1)*(n+2) else 3*Self(n-1) - 2*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Apr 19 2021

LinearRecurrence[{3,-2,1}, {3,8,20}, 40] (* G. C. Greubel, Apr 19 2021 *)
CoefficientList[Series[(3-x+2x^2)/(1-3x+2x^2-x^3),{x,0,40}],x] (* Harvey P. Dale, Oct 15 2021 *)

@CachedFunction
def a(n): return 2^n*(n+3) if n<3 else sum((-1)^j*(3-j)*a(n-j-1) for j in (0..2))
[a(n) for n in (0..40)] # G. C. Greubel, Apr 19 2021

G.f.: f(z) = 3 +8*z + ... = (3 -z +2*z^2)/(1 -3*z +2*z^2 -z^3).

a(n+3) = 3*a(n+2) -2*a(n+1) +a(n) (n>=0). - Richard Choulet, Apr 07 2009

A098601 Expansion of (1+2x)/((1+x)(1-x^2-x^3)).

Original entry on oeis.org

1, 1, 0, 3, 0, 4, 2, 5, 5, 8, 9, 14, 16, 24, 29, 41, 52, 71, 92, 124, 162, 217, 285, 380, 501, 666, 880, 1168, 1545, 2049, 2712, 3595, 4760, 6308, 8354, 11069, 14661, 19424, 25729, 34086, 45152, 59816, 79237, 104969, 139052, 184207, 244020, 323260

Offset: 0

Paul Barry, Sep 17 2004

Diagonal sums of A098599.

The signed sequence 1,-1,0,-3,0,-4,... gives the diagonal sums of A100218. - Paul Barry, Nov 09 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,1,2,1).

Cf. A000931, A077883, A098599, A100218, A135364.

I:=[1,1,0,3]; [n le 4 select I[n] else -Self(n-1) +Self(n-2) +2*Self(n-3) +Self(n-4): n in [1..55]]; // G. C. Greubel, Mar 27 2024

CoefficientList[Series[(1+2x)/((1+x)(1-x^2-x^3)),{x,0,50}],x] (* or *) LinearRecurrence[{-1,1,2,1},{1,1,0,3},50] (* Harvey P. Dale, Dec 14 2011 *)

def A098601(n): return sum( binomial(k, n-2*k) + binomial(k-1, n-2*k-1) for k in range(1+n//2))
[A098601(n) for n in range(56)] # G. C. Greubel, Mar 27 2024

G.f.: x/((1+x)*(1-x^2-x^3)) + 1/(1-x^2-x^3).
a(n) = Sum_{k=0..floor(n/2)} (binomial(k, n-2*k) + binomial(k-1, n-2*k-1)).
a(n) = -a(n-1) + a(n-2) + 2*a(n-3) + a(n-4).
Inverse binomial transform of A135364. - Paul Curtz, Apr 25 2008

A159340 Transform of the finite sequence (1, 0, -1) by the T_{0,1} transformation (see link).

Original entry on oeis.org

2, 3, 6, 16, 38, 88, 204, 474, 1102, 2562, 5956, 13846, 32188, 74828, 173954, 404394, 940102, 2185472, 5080606, 11810976, 27457188, 63830218, 148387254, 344958514, 801931252, 1864263982, 4333887956, 10075067156, 23421689538, 54448822258

Offset: 0

Richard Choulet, Apr 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Cf. A135364.

I:=[6, 16, 38]; [2, 3] cat [n le 3 select I[n] else 3*Self(n-1) - 2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 25 2018

a(0):=2: a(1):=3:a(2):=6: a(3):=16:a(4):=38:for n from 2 to 31 do a(n+3):=3*a(n+2)-2*a(n+1)+a(n):od:seq(a(i),i=0..31);

Join[{2, 3}, LinearRecurrence[{3, -2, 1}, {6, 16, 38}, 49]] (* G. C. Greubel, Jun 25 2018 *)

z='z+O('z^30); Vec(((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2)+((1-z+z^2)/(1-3*z+2*z^2-z^3))) \\ G. C. Greubel, Jun 25 2018

O.g.f.: f(z) = ((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2)+((1-z+z^2)/(1-3*z+2*z^2-z^3)).

a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) for n >= 5, with a(0)=2, a(1)=3, a(2)=6, a(3)=16, a(4)=38.

A159341 Transform of the finite sequence (1, 0, -1, 0, 1) by the T_{0,1} transformation (see link).

Original entry on oeis.org

2, 3, 6, 16, 39, 89, 206, 479, 1114, 2590, 6021, 13997, 32539, 75644, 175851, 408804, 950354, 2209305, 5136011, 11939777, 27756614, 64526299, 150005446, 348720354, 810676469, 1884594145, 4381149851, 10184937732, 23677107639, 55042597304

Offset: 0

Richard Choulet, Apr 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Cf. A135364, A159340.

I:=[39, 89, 206]; [2, 3, 6, 16] cat [n le 3 select I[n] else 3*Self(n-1) - 2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 25 2018

a(0):=2: a(1):=3:a(2):=6: a(3):=16:a(4):=39:a(5):=89:a(6):=206:for n from 4 to 31 do a(n+3):=3*a(n+2)-2*a(n+1)+a(n):od:seq(a(i),i=0..31);

Join[{2, 3, 6, 16}, LinearRecurrence[{3, -2, 1}, {39, 89, 206}, 47]] (* G. C. Greubel, Jun 25 2018 *)

z='z+O('z^30); Vec(((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2+z^4) + ((1-z+z^2)/(1-3*z+2*z^2-z^3))) \\ G. C. Greubel, Jun 25 2018

O.g.f: f(z) = ((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2+z^4) + ((1-z+z^2)/(1-3*z+2*z^2-z^3)).

a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) for n >= 7, with a(0)=2, a(1)=3, a(2)=6, a(3)=16, a(4)=39, a(5)=89, a(6)=206.

A159342 Transform of the finite sequence (1, 0, -1, 0, 1, 0, -1) by the T_{0,1} transform (see link).

Original entry on oeis.org

2, 3, 6, 16, 39, 89, 207, 480, 1116, 2595, 6033, 14025, 32604, 75795, 176202, 409620, 952251, 2213715, 5146263, 11963610, 27812019, 64655100, 150304872, 349416435, 812294661, 1888355985, 4389895068, 10205267895, 23724369534, 55152467880

Offset: 0

Richard Choulet, Apr 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Cf. A135364, A159340, A159341.

I:=[207, 480, 1116]; [2, 3, 6, 16, 39, 89] cat [n le 3 select I[n] else 3*Self(n-1) - 2*Self(n-2) +Self(n-3): n in [1..50]]; // G. C. Greubel, Jun 17 2018

a(0):=2: a(1):=3:a(2):=6: a(3):=16:a(4):=39:a(5):=89:a(6):=207:a(7):=480:a(8):=1116:for n from 6 to 31 do a(n+3):=3*a(n+2)-2*a(n+1)+a(n):od:seq(a(i),i=0..31);

Join[{2, 3, 6, 16, 39, 89}, LinearRecurrence[{3, -2, 1}, {207, 480, 1116}, 50]] (* G. C. Greubel, Jun 17 2018 *)

m=50; v=concat([207, 480, 1116], vector(m-3)); for(n=4, m, v[n] = 3*v[n-1] -2*v[n-2] +v[n-3]); concat([2, 3, 6, 16, 39, 89], v) \\ G. C. Greubel, Jun 17 2018

O.g.f.: ((1-x)^2/(1-3*x+2*x^2-x^3))*(1-x^2+x^4+x^6)+((1-x+x^2)/(1-3*x+2*x^2-x^3)).

a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) for n >= 9, with a(0)=2, a(1)=3, a(2)=6, a(3)=16, a(4)=39, a(5)=89, a(6)=207, a(7)=480, a(8)=1116.

A159343 Transform of A056594 by the T_{0,1} transformation (see link).

Original entry on oeis.org

2, 3, 6, 16, 39, 89, 205, 478, 1113, 2586, 6010, 13973, 32485, 75517, 175554, 408115, 948754, 2205584, 5127359, 11919665, 27709861, 64417610, 149752773, 348132962, 809310950, 1881419697, 4373770153, 10167782017, 23637225442, 54949882443

Offset: 0

Richard Choulet, Apr 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..2500
Richard Choulet, Curtz-like transformation.
Index entries for linear recurrences with constant coefficients, signature (3,-3,4,-2,1).

Cf. A135364, A159340, A159441, A159442.

I:=[2, 3, 6, 16, 39]; [n le 5 select I[n] else 3*Self(n-1) -3*Self(n-2) +4*Self(n-3) -2*Self(n-4) +Self(n-5): n in [1..50]]; // G. C. Greubel, Jun 17 2018

a(0):=2: a(1):=3:a(2):=6: a(3):=16:a(4):=39:for n from 0 to 31 do a(n+5):=3*a(n+4)-3*a(n+3)+4*a(n+2)-2*a(n+1)+a(n):od:seq(a(i),i=0..31);

LinearRecurrence[{3, -3, 4, -2, 1}, {2, 3, 6, 16, 39}, 50] (* G. C. Greubel, Jun 17 2018 *)

m=32; v=concat([2, 3, 6, 16, 39], vector(m-5)); for(n=6, m, v[n] = 3*v[n-1] -3*v[n-2] +4*v[n-3] -2*v[n-4] +v[n-5]); v \\ G. C. Greubel, Jun 17 2018

O.g.f.: ((1-z)^2/(1-3*z+2*z^2-z^3))*(1/(1+z^2))+((1-z+z^2)/(1-3*z+2*z^2-z^3)).

a(n) = 3*a(n-1) - 3*a(n-2) + 4*a(n-3) - 2*a(n-4) + a(n-5) for n>=5, with a(0)=2, a(1)=3, a(2)=6, a(3)=16, a(4)=39.

A137495 a(n) = A098601(2n) + A098601(2n+1).

Original entry on oeis.org

2, 3, 4, 7, 13, 23, 40, 70, 123, 216, 379, 665, 1167, 2048, 3594, 6307, 11068, 19423, 34085, 59815, 104968, 184206, 323259, 567280, 995507, 1746993, 3065759, 5380032, 9441298, 16568323, 29075380, 51023735, 89540413, 157132471, 275748264, 483904470, 849193147, 1490230088

Offset: 0

Paul Curtz, Apr 27 2008

Nicolas Ollinger and Jeffrey Shallit, The Repetition Threshold for Rote Sequences, arXiv:2406.17867 [math.CO], 2024.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1)

Cf. A005314, A135364, A137584, A137531.

LinearRecurrence[{2, -1, 1}, {2, 3, 4}, 50] (* Paolo Xausa, Jun 29 2024 *)

Vec((x-2)/(-1+2*x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Jul 06 2011

a(3n) = A135364(2n+1). a(3n+1) = A137584(2n+1). a(3n+2) = A137531(2n+2).
From R. J. Mathar, Jul 06 2011: (Start)
G.f.: ( -2+x ) / ( -1+2*x-x^2+x^3 ).
a(n) = 2*A005314(n+1) - A005314(n). (End)

cp's OEIS Frontend

A097550 Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.

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A136303 Expansion of g.f. (1 +x^2)/((1-x)^2*(1 -3*x +2*x^2 -x^3)).

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A136304 Expansion of g.f. (1-z)^2*(1-sqrt(1-4*z))/(2*z*(1 - 3*z + 2*z^2 - z^3)).

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A136305 Expansion of g.f. (3 -x +2*x^2)/(1 -3*x +2*x^2 -x^3).

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A098601 Expansion of (1+2*x)/((1+x)*(1-x^2-x^3)).

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A159340 Transform of the finite sequence (1, 0, -1) by the T_{0,1} transformation (see link).

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A159341 Transform of the finite sequence (1, 0, -1, 0, 1) by the T_{0,1} transformation (see link).

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A136303 Expansion of g.f. (1 +x^2)/((1-x)^2(1 -3x +2*x^2 -x^3)).

A136304 Expansion of g.f. (1-z)^2(1-sqrt(1-4z))/(2z(1 - 3z + 2z^2 - z^3)).

A136305 Expansion of g.f. (3 -x +2x^2)/(1 -3x +2*x^2 -x^3).

A098601 Expansion of (1+2x)/((1+x)(1-x^2-x^3)).