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 1-6 of 6 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

A097551 Number of positive words of length n in the monoid Br_4 of positive braids on 5 strands.

Original entry on oeis.org

1, 4, 13, 37, 101, 273, 737, 1990, 5374, 14513, 39194, 105848, 285855, 771985, 2084836, 5630344, 15205404, 41063976, 110898081, 299493268, 808816679, 2184304257, 5898969706, 15930859211, 43023152830, 116189067703
Offset: 0

Views

Author

D n Verma, Aug 16 2004

Keywords

Links

Crossrefs

Cf. A097550, A097552, A097553, A097554, A097555, A097556.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( (1+x^2)^2/(1-4*x+5*x^2-5*x^3+3*x^4-x^5) )); // G. C. Greubel, Apr 19 2021
  • Mathematica
    <Ryan Propper, Sep 27 2005 *)
LinearRecurrence[{4,-5,5,-3,1}, {1,4,13,37,101}, 51] (* G. C. Greubel, Apr 19 2021 *)
  • Sage
    def A097551_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^2/(1-4*x+5*x^2-5*x^3+3*x^4-x^5) ).list()
A097551_list(50) # G. C. Greubel, Apr 19 2021

Formula

G.f.: (1 + x^2)^2/(1 - 4*x + 5*x^2 - 5*x^3 + 3*x^4 - x^5).

Extensions

More terms from Ryan Propper, Sep 27 2005

A097552 Number of positive words of length n in the monoid Br_5 of positive braids on 6 strands.

Original entry on oeis.org

1, 5, 20, 67, 209, 630, 1873, 5540, 16357, 48265, 142387, 420027, 1239006, 3654820, 10780958, 31801551, 93807834, 276713194, 816245143, 2407749755, 7102350204, 20950424039, 61799299470, 182294802589, 537730934397
Offset: 0

Views

Author

D n Verma, Aug 16 2004

Keywords

Links

Crossrefs

Cf. A097550, A097551, A097553, A097554, A097555, A097556.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x^2)^3/(1-5*x+8*x^2-7*x^3+4*x^4-x^5) )); // G. C. Greubel, Apr 19 2021
  • Mathematica
    LinearRecurrence[{5,-8,7,-4,1}, {1,5,20,67,209,630,1873}, 40] (* G. C. Greubel, Apr 19 2021 *)
  • Sage
    def A097552_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^3/(1-5*x+8*x^2-7*x^3+4*x^4-x^5) ).list()
A097552_list(40) # G. C. Greubel, Apr 19 2021

Formula

G.f.: (1 + x^2)^3/(1 - 5*x + 8*x^2 - 7*x^3 + 4*x^4 - x^5). - T. D. Noe, Nov 02 2006

Extensions

Corrected by T. D. Noe, Nov 02 2006

A097553 Number of positive words of length n in the monoid Br_6 of positive braids on 7 strands.

Original entry on oeis.org

1, 6, 27, 101, 346, 1131, 3611, 11396, 35761, 111906, 349700, 1092039, 3409031, 10640179, 33206991, 103631414, 323402952, 1009233980, 3149469548, 9828376731, 30670834516, 95712596642, 298684343689, 932085486213, 2908700435744
Offset: 0

Views

Author

D n Verma, Aug 16 2004

Keywords

Links

Crossrefs

Cf. A097550, A097551, A097552, A097554, A097555, A097556.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( (1+x^2)^4/(1-6*x+13*x^2-17*x^3+17*x^4-11*x^5+5*x^6-x^7) )); // G. C. Greubel, Apr 20 2021
  • Mathematica
    CoefficientList[Series[(1+n^2)^4/(1-6n+13n^2-17n^3+17n^4-11n^5+5n^6-n^7),{n,0,30}],n] (* Harvey P. Dale, Sep 27 2019 *)
LinearRecurrence[{6,-13,17,-17,11,-5,1}, {1,6,27,101,346,1131,3611,11396,35761}, 40] (* G. C. Greubel, Apr 20 2021 *)
  • Sage
    def A097553_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^4/(1-6*x+13*x^2-17*x^3+17*x^4-11*x^5+5*x^6-x^7) ).list()
A097553_list(50) # G. C. Greubel, Apr 20 2021

Formula

G.f.: (1 +x^2)^4/(1 -6*x +13*x^2 -17*x^3 +17*x^4 -11*x^5 +5*x^6 -x^7).

Extensions

Corrected and extended by Max Alekseyev, Jun 17 2011

A097554 Number of positive words of length n in the monoid Br_7 of positive braids on 8 strands.

Original entry on oeis.org

1, 7, 36, 151, 570, 2019, 6893, 23034, 76020, 249077, 812614, 2644447, 8592693, 27895296, 90510106, 293576779, 952053411, 3087093728, 10009389358, 32452403488, 105214363653, 341111617862, 1105895184121, 3585328906357, 11623651559099
Offset: 0

Views

Author

D n Verma, Aug 16 2004

Keywords

Links

Crossrefs

Cf. A097550, A097551, A097552, A097553, A097555, A097556.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x^2)^5/(1-7*x+18*x^2-25*x^3+24*x^4-15*x^5+6*x^6-x^7) )); // G. C. Greubel, Apr 20 2021
  • Mathematica
    LinearRecurrence[{7,-18,25,-24,15,-6,1}, {1,7,36,151,570,2019,6893,23034,76020, 249077,812614}, 41] (* G. C. Greubel, Apr 20 2021 *)
  • Sage
    def A097554_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^5/(1-7*x+18*x^2-25*x^3+24*x^4-15*x^5+6*x^6-x^7) ).list()
A097554_list(40) # G. C. Greubel, Apr 20 2021

Formula

G.f.: (1 +x^2)^5/(1 -7*x +18*x^2 -25*x^3 +24*x^4 -15*x^5 +6*x^6 -x^7).

Extensions

Corrected and extended by Max Alekseyev, Jun 17 2011

A097556 Number of positive words of length n in the monoid Br_9 of positive braids on 10 strands.

Original entry on oeis.org

1, 9, 56, 279, 1223, 4932, 18833, 69345, 249166, 880525, 3076295, 10662459, 36749785, 126161246, 431880044, 1475412473, 5032964258, 17150277106, 58395929325, 198723871661, 675989712225, 2298799014859, 7815699898677, 26568450635871
Offset: 0

Views

Author

D n Verma, Aug 16 2004

Keywords

Links

Crossrefs

Cf. A097550, A097551, A097552, A097553, A097554, A097555.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 50);
Coefficients(R!( (1+x^2)^7/(1-9*x+32*x^2-63*x^3+84*x^4-81*x^5+56*x^6-27*x^7+8*x^8-x^9) )); // G. C. Greubel, Apr 20 2021
  • Mathematica
    CoefficientList[Series[(1+x^2)^7/(1-9*x+32*x^2-63*x^3+84*x^4-81*x^5+56*x^6-27*x^7+8*x^8-x^9), {x,0,50}], x] (* G. C. Greubel, Apr 20 2021 *)
  • Sage
    def A097556_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)^7/(1-9*x+32*x^2-63*x^3+84*x^4-81*x^5+56*x^6-27*x^7+8*x^8-x^9) ).list()
A097556_list(50) # G. C. Greubel, Apr 20 2021

Formula

G.f.: (1 +x^2)^7/(1 -9*x +32*x^2 -63*x^3 +84*x^4 -81*x^5 +56*x^6 -27*x^7 +8*x^8 -x^9).

Extensions

Edited and extended by Max Alekseyev, Jun 17 2011
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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