cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Original entry on oeis.org

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

Views

Author

D n Verma, Aug 16 2004

Keywords

Links

Crossrefs

Cf. A097551, A097552, A097553, A097554, A097555, A097556.
Cf. A095263, A135364, A136302, A136303, A136304, A136305, A137229, A137234, A137249.

Programs

  • Magma
    [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
  • Maple
    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
  • Mathematica
    LinearRecurrence[{3,-2,1},{1,3,8},30] (* Harvey P. Dale, Jul 10 2019 *)
  • Sage
    @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

Formula

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

Extensions

More terms from Ryan Propper, Sep 27 2005

A135364 First column of a triangle - see Comments lines.

Original entry on oeis.org

1, 2, 3, 7, 17, 40, 93, 216, 502, 1167, 2713, 6307, 14662, 34085, 79238, 184206, 428227, 995507, 2314273, 5380032, 12507057, 29075380, 67592058, 157132471, 365288677, 849193147, 1974134558, 4589306057, 10668842202
Offset: 0

Views

Author

Paul Curtz, Dec 09 2007

Keywords

Comments

...1;
...2,...1;
...3,...3,...1;
...7,...5,...4,...1;
..17,..10,...7,...5,...1;
..40,..24,..13,...9,...6,...1;
..93,..57,..31,..16,..11,...7,...1;
From the second, the sum of a row gives the first term of the following one. Diagonal differences are the first term upon. First column is a(n).

Links

Crossrefs

Cf. A034943, A097550, A136302, A136303, A136304, A136305, A137229, A137234, A137249.

Programs

  • Magma
    I:=[3,7,17]; [1,2] cat [n le 3 select I[n] else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..51]]; // G. C. Greubel, Apr 19 2021
  • Maple
    a:= n-> `if`(n=0, 1, (<<7|3|2>> .<<3|1|0>, <-2|0|1>, <1|0|0>>^(n-1))[1, 3]):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 24 2008
  • Mathematica
    LinearRecurrence[{3,-2,1}, {1,2,3,7,17}, 51] (* G. C. Greubel, Oct 11 2016; Apr 19 2021 *)
  • Sage
    @CachedFunction
def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) )
def A135364(n): return 1 if n==0 else 2*A095263(n-1) -3*A095263(n-2) +2*A095263(n-3)
[A135364(n) for n in (0..50)] # G. C. Greubel, Apr 19 2021

Formula

From Richard Choulet, Jan 06 2008: (Start)
a(n+1) = a(n) + a(n-1) + (n-1)*a(1) + (n-2)*a(2) + ... + 2*a(n-2) for n>=3.
O.g.f.: 1 + x*(2 - 3*x + 2*x^2) / (1 - 3*x + 2*x^2 - x^3).
a(n+3) = 3*a(n+2) - 2*a(n+1) + a(n). (End)
a(n) = A034943(n) + A034943(n+1). - R. J. Mathar, Apr 09 2008
a(0) = 1, a(n) = term (1,3) in the 1 X 3 matrix [7,3,2].[3,1,0; -2,0,1; 1,0,0]^(n-1) (n>0). - Alois P. Heinz, Jul 24 2008
a(n) = 2*A095263(n-1) -3*A095263(n-2) +2*A095263(n-3) with a(0) = 1. - G. C. Greubel, Apr 19 2021

Extensions

More terms from Richard Choulet, Jan 06 2008

A136302 Transform of A000027 by the T_{1,1} transformation (see link).

Original entry on oeis.org

2, 6, 15, 35, 81, 188, 437, 1016, 2362, 5491, 12765, 29675, 68986, 160373, 372822, 866706, 2014847, 4683951, 10888865, 25313540, 58846841, 136802308, 318026782, 739322571, 1718716457, 3995531011, 9288482690, 21593102505, 50197873146, 116695897118, 271285047567
Offset: 1

Views

Author

Richard Choulet, Mar 22 2008

Keywords

Links

Crossrefs

Cf. A095263, A136303, A136304, A136305.

Programs

  • Magma
    I:=[2,6,15]; [n le 3 select I[n] else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..41]]; // G. C. Greubel, Apr 12 2021
  • Maple
    a:= n-> (<<6|2|1>>. <<3|1|0>, <-2|0|1>, <1|0|0>>^n)[1, 3]:
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 14 2008
  • Mathematica
    LinearRecurrence[{3,-2,1}, {2,6,15}, 41] (* G. C. Greubel, Apr 12 2021 *)
  • Sage
    def A136302_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(2+x^2)/(1-3*x+2*x^2-x^3) ).list()
a=A136302_list(41); a[1:] # G. C. Greubel, Apr 12 2021

Formula

G.f.: 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
a(n) = 2*A095263(n) + A095263(n-2). - R. J. Mathar, Feb 29 2016

Extensions

More terms from Alois P. Heinz, Aug 14 2008

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

Views

Author

Richard Choulet, Mar 22 2008

Keywords

Comments

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

Links

Crossrefs

Cf. A097550, A135364, A136302, A136303, A136305, A137229, A137234, A137249.
Cf. A000108, A034943.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • Sage
    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

Formula

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

Extensions

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

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

Views

Author

Richard Choulet, Mar 22 2008

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Magma
    [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
  • Mathematica
    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 *)
  • Sage
    @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

Formula

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

A137229 Expansion of g.f. x/((1-x)*(1-3*x+2*x^2-x^3)).

Original entry on oeis.org

1, 4, 11, 27, 64, 150, 350, 815, 1896, 4409, 10251, 23832, 55404, 128800, 299425, 696080, 1618191, 3761839, 8745216, 20330162, 47261894, 109870575, 255418100, 593775045, 1380359511, 3208946544, 7459895656, 17342153392, 40315615409, 93722435100, 217878227875
Offset: 1

Views

Author

Richard Choulet, Apr 05 2008

Keywords

Comments

Previous name was: Transform of A000217 without the initial 0 by the T_{0,0} transformation (see link).
Partial sums of A095263. - R. J. Mathar, Nov 04 2008

Links

Crossrefs

Cf. A136302, A136303, A136304, A136305.

Programs

  • Magma
    I:=[1,4,11,27]; [n le 4 select I[n] else 4*Self(n-1) -5*Self(n-2) +3*Self(n-3) -Self(n-4): n in [1..40]]; // G. C. Greubel, Apr 17 2021
  • Maple
    a:= n-> (<<3|1|0|0>, <-2|0|1|0>, <1|0|0|0>, <1|0|0|1>>^n)[4, 1]:
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 24 2008
  • Mathematica
    LinearRecurrence[{4,-5,3,-1},{1,4,11,27},40] (* Harvey P. Dale, Nov 10 2014 *)
  • Sage
    def A137229_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x/((1-x)*(1-3*x+2*x^2-x^3)) ).list()
a=A137229_list(41); a[1:] # G. C. Greubel, Apr 17 2021

Formula

O.g.f: x/((1-x)*(1 -3*x +2*x^2 -x^3)).
a(n) = term (4,1) in the 4x4 matrix [3,1,0,0; -2,0,1,0; 1,0,0,0; 1,0,0,1]^(n). - Alois P. Heinz, Jul 24 2008

Extensions

New name using g.f., Joerg Arndt, Apr 18 2021

A137234 Expansion of g.f. 1/((1-x)^2*(1 - 3*x + 2*x^2 - x^3)).

Original entry on oeis.org

1, 5, 16, 43, 107, 257, 607, 1422, 3318, 7727, 17978, 41810, 97214, 226014, 525439, 1221519, 2839710, 6601549, 15346765, 35676927, 82938821, 192809396, 448227496, 1042002541, 2422362052, 5631308596, 13091204252, 30433357644, 70748973053
Offset: 0

Views

Author

Richard Choulet, Apr 05 2008

Keywords

Comments

Previous name: Transform of A000292 without the initial 0 by the T_{0,0} transformation (see link).
Partial sums of A137229. - R. J. Mathar, Nov 04 2008

Links

Crossrefs

Cf. A095263, A136302, A136303, A136304, A136305.

Programs

  • Magma
    I:=[1,5,16,43,107]; [n le 5 select I[n] else 5*Self(n-1) -9*Self(n-2) +8*Self(n-3) -4*Self(n-4) +Self(n-5): n in [1..41]]; // G. C. Greubel, Apr 19 2021
  • Mathematica
    LinearRecurrence[{5,-9,8,-4,1}, {1,5,16,43,107}, 41] (* G. C. Greubel, Apr 19 2021 *)
CoefficientList[Series[1/((1-x)^2(1-3x+2x^2-x^3)),{x,0,30}],x] (* Harvey P. Dale, Jun 07 2021 *)
  • Sage
    @CachedFunction
def A095263(n): return sum(binomial(n+j+2, 3*j+2) for j in (0..n//2))
def A137234(n): return -(n+3) + sum( (-1)^j*(4-j)*A095263(n-j) for j in (0..2))
[A137234(n) for n in (0..40)] # G. C. Greubel, Apr 19 2021

Formula

O.g.f: 1/((1-z)^2*(1 - 3*z + 2*z^2 - z^3)).
a(n) = -(n+3) + Sum_{j=0..2} (-1)^j*(4-j)*A095263(n-j). - G. C. Greubel, Apr 19 2021

A137249 Expansion of g.f. z*(2-2*z+z^2+z^3)/((1+z)*(1-3*z+2*z^2-z^3)).

Original entry on oeis.org

2, 2, 7, 15, 37, 84, 197, 456, 1062, 2467, 5737, 13335, 31002, 72069, 167542, 389486, 905447, 2104907, 4893317, 11375580, 26445017, 61477204, 142917162, 332242091, 772369157, 1795540447, 4174125122, 9703663625, 22558281082
Offset: 1

Views

Author

Richard Choulet, Apr 05 2008

Keywords

Comments

Previous name was: Transform of A033999 by the T_{0,1} transformation (see link).

Links

Crossrefs

Cf. A095263, A136302, A136303, A136304, A136305.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (2-2*x+x^2+x^3)/((1+x)*(1-3*x+2*x^2-x^3)) )); // G. C. Greubel, Apr 11 2021
  • Maple
    m:= 40;
S:= series( x*(2-2*x+x^2+x^3)/((1+x)*(1-3*x+2*x^2-x^3)), x, m+1);
seq(coeff(S, x, j), j = 1..m); # G. C. Greubel, Apr 11 2021
  • Mathematica
    LinearRecurrence[{2,1,-1,1},{2,2,7,15},30] (* Harvey P. Dale, Feb 02 2012 *)
  • Sage
    def A132749_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (2-2*x+x^2+x^3)/((1+x)*(1-3*x+2*x^2-x^3)) ).list()
A132749_list(40) # G. C. Greubel, Apr 11 2021

Formula

O.g.f: z*(2 -2*z +z^2 +z^3)/( (1+z)*(1-3*z+2*z^2-z^3) ).
a(n+4) = 2*a(n+3) + a(n+2) - a(n+1) + a(n).
From G. C. Greubel, Apr 11 2021: (Start)
a(n) = (4*(-1)^n + 10*A095263(n) - 12*A095263(n-1) + 11*A095263(n-2))/7.
a(n) = (1/7)*( 4*(-1)^n + Sum_{j=0..floor(n/2)} ( 10*binomial(n+j+2, 3*j+2) - 12*binomial(n+j+1, 3*j+2) + 11*binomial(n+j, 3*j+2) ) ). (End)

Extensions

New name using g.f. from Joerg Arndt, Apr 19 2021
Showing 1-8 of 8 results.

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