A170745 -id:A170745

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A003945

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

+10
212

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Coordination sequence for infinite tree with valency 3.

Number of Hamiltonian cycles in K_3 X P_n.

Number of ternary words of length n avoiding aa, bb, cc.

For n > 0, row sums of A029635. - Paul Barry, Jan 30 2005

Binomial transform is {1, 4, 13, 40, 121, 364, ...}, see A003462. - Philippe Deléham, Jul 23 2005

Convolved with the Jacobsthal sequence A001045 = A001786: (1, 4, 12, 32, 80, ...). - Gary W. Adamson, May 23 2009

Equals (n+1)-th row sums of triangle A161175. - Gary W. Adamson, Jun 05 2009

a(n) written in base 2: a(0) = 1, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009

INVERTi transform of A003688. - Gary W. Adamson, Aug 05 2010

An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 42, 138, 162 and 168, lead to this sequence. For the corner squares these vectors lead to the companion sequence A083329. - Johannes W. Meijer, Aug 15 2010

A216022(a(n)) != 2 and A216059(a(n)) != 3. - Reinhard Zumkeller, Sep 01 2012

Number of length-n strings of 3 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012

Sums of pairs of rows of Pascal's triangle A007318, T(2n,k)+T(2n+1,k); Sum_{n>=1} A000290(n)/a(n) = 4. - John Molokach, Sep 26 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math. (2024) Vol. 79, 173.

F. Faase, Counting Hamiltonian cycles in product graphs

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 151

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 304

Markus Kuba and Alois Panholzer, Enumeration formulas for pattern restricted Stirling permutations, Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From N. J. A. Sloane, Sep 25 2012

C. Richard and U. Grimm, On the entropy and letter frequencies of ternary squarefree words, arXiv:math/0302302 [math.CO], 2003.

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (2).

Index entries for sequences related to trees

FORMULA

a(0) = 1; for n > 0, a(n) = 3*2^(n-1).

a(n) = 2*a(n-1), n > 1; a(0)=1, a(1)=3.

More generally, the g.f. (1+x)/(1-k*x) produces the sequence [1, 1 + k, (1 + k)*k, (1 + k)*k^2, ..., (1+k)*k^(n-1), ...], with a(0) = 1, a(n) = (1+k)*k^(n-1) for n >= 1. Also a(n+1) = k*a(n) for n >= 1. - Zak Seidov and N. J. A. Sloane, Dec 05 2009

The g.f. (1+x)/(1-k*x) produces the sequence with closed form (in PARI notation) a(n)=(n>=0)*k^n+(n>=1)*k^(n-1). - Jaume Oliver Lafont, Dec 05 2009

Binomial transform of A000034. a(n) = (3*2^n - 0^n)/2. - Paul Barry, Apr 29 2003

a(n) = Sum_{k=0..n} (n+k)*binomial(n, k)/n. - Paul Barry, Jan 30 2005

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 1. - Philippe Deléham, Jul 10 2005

Binomial transform of A000034. Hankel transform is {1,-3,0,0,0,...}. - Paul Barry, Aug 29 2006

a(0) = 1, a(n) = 2 + Sum_{k=0..n-1} a(k) for n >= 1. - Joerg Arndt, Aug 15 2012

a(n) = 2^n + floor(2^(n-1)). - Martin Grymel, Oct 17 2012

E.g.f.: (3*exp(2*x) - 1)/2. - Stefano Spezia, Jan 31 2023

MAPLE

k := 3; if n = 0 then 1 else k*(k-1)^(n-1); fi;

MATHEMATICA

Join[{1}, 3*2^Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

Table[2^n+Floor[2^(n-1)], {n, 0, 30}] (* Martin Grymel, Oct 17 2012 *)

CoefficientList[Series[(1+x)/(1-2x), {x, 0, 40}], x] (* or *) LinearRecurrence[ {2}, {1, 3}, 40] (* Harvey P. Dale, May 04 2017 *)

PROG

(PARI) a(n)=if(n, 3<<n--, 1) \\ Charles R Greathouse IV, Jan 12 2012

CROSSREFS

Essentially same as A007283 (3*2^n) and A042950.

Cf. A001787, A001045, A161175.

Generating functions of the form (1+x)/(1-k*x) for k=1 to 12: A040000, A003945, A003946, A003947, A003948, A003949, A003950, A003951, A003952.

Generating functions of the form (1+x)/(1-k*x) for k=13 to 30: A170732, A170733, A170734, A170735, A170736, A170737, A170738, A170739, A170740, A170741, A170742, A170743, A170744, A170745, A170746, A170747, A170748.

Generating functions of the form (1+x)/(1-k*x) for k=31 to 50: A170749, A170750, A170751, A170752, A170753, A170754, A170755, A170756, A170757, A170758, A170759, A170760, A170761, A170762, A170763, A170764, A170765, A170766, A170767, A170768, A170769.

Cf. A003688.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by N. J. A. Sloane, Dec 04 2009

STATUS

approved

A163526

Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

+10
1

1, 26, 650, 16250, 406250, 10155925, 253890000, 6347047200, 158671110000, 3966651000000, 99163106355300, 2478998445300000, 61972980856207200, 1549275016079700000, 38730637808401500000, 968235006358878382800

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170745, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..710

Index entries for linear recurrences with constant coefficients, signature (24, 24, 24, 24, -300).

FORMULA

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

a(n) = 24*a(n-1)+24*a(n-2)+24*a(n-3)+24*a(n-4)-300*a(n-5). - Wesley Ivan Hurt, May 10 2021

MATHEMATICA

CoefficientList[Series[(1+x)*(1-x^5)/(1-25*x+324*x^5-300*x^6), {x, 0, 20}], x] (* G. C. Greubel, Jul 27 2017 *)

coxG[{5, 300, -24}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 16 2019 *)

PROG

(PARI) my(x='x+O('x^20)); Vec((1+x)*(1-x^5)/(1-25*x+324*x^5-300*x^6)) \\ G. C. Greubel, Jul 27 2017

(Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^5)/(1-25*x+324*x^5-300*x^6) )); // G. C. Greubel, May 16 2019

(Sage) ((1+x)*(1-x^5)/(1-25*x+324*x^5-300*x^6)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, May 16 2019

KEYWORD

nonn

AUTHOR

John Cannon and N. J. A. Sloane, Dec 03 2009

STATUS

approved

A163995

Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

+10
1

1, 26, 650, 16250, 406250, 10156250, 253905925, 6347640000, 158690797200, 3967264860000, 99181494750000, 2479534200000000, 61988275781355300, 1549704914070300000, 38742573340231207200, 968563095719204700000, 24214046448355276500000, 605350387594249537500000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170745, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..710

Index entries for linear recurrences with constant coefficients, signature (24,24,24,24,24,-300).

FORMULA

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

a(n) = -300*a(n-6) + 24*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021

MAPLE

seq(coeff(series((1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019

MATHEMATICA

CoefficientList[Series[(1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7), {t, 0, 30}], t] (* G. C. Greubel, Aug 24 2017 *)

coxG[{6, 300, -24}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 13 2019 *)

PROG

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7)) \\ G. C. Greubel, Aug 24 2017

(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7) )); // G. C. Greubel, Aug 13 2019

(Sage)

def A163995_list(prec):

P.<t> = PowerSeriesRing(ZZ, prec)

return P((1+t)*(1-t^6)/(1-25*t+324*t^6-300*t^7)).list()

A163995_list(30) # G. C. Greubel, Aug 13 2019

(GAP) a:=[26, 650, 16250, 406250, 10156250, 253905925];; for n in [7..30] do a[n]:=24*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -300*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 13 2019

KEYWORD

nonn

AUTHOR

John Cannon and N. J. A. Sloane, Dec 03 2009

STATUS

approved

A165973

Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

+10
1

1, 26, 650, 16250, 406250, 10156250, 253906250, 6347656250, 158691406250, 3967285156250, 99182128905925, 2479553222640000, 61988830565797200, 1549720764139860000, 38743019103369750000, 968575477581075000000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170745, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (24,24,24,24,24,24,24,24,24,-300).

FORMULA

G.f.: (t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

MAPLE

seq(coeff(series((1+t)*(1-t^10)/(1-25*t+324*t^10-300*t^11), t, n+1), t, n), n = 0..30); # G. C. Greubel, Sep 26 2019

MATHEMATICA

coxG[{10, 300, -24}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Mar 03 2016 *)

CoefficientList[Series[(1+t)*(1-t^10)/(1-25*t+324*t^10-300*t^11), {t, 0, 25}], t] (* G. C. Greubel, Sep 26 2019 *)

PROG

(PARI) my(t='t+O('t^30)); Vec((1+t)*(1-t^10)/(1-25*t+324*t^10-300*t^11)) \\ G. C. Greubel, Sep 26 2019

(Magma) R<t>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^10)/(1-25*t+324*t^10-300*t^11) )); // G. C. Greubel, Sep 26 2019

(Sage)

def A165973_list(prec):

P.<t> = PowerSeriesRing(ZZ, prec)

return P((1+t)*(1-t^10)/(1-25*t+324*t^10-300*t^11)).list()

A165973_list(30) # G. C. Greubel, Sep 26 2019

(GAP) a:=[26, 650, 16250, 406250, 10156250, 253906250, 6347656250, 158691406250, 3967285156250, 99182128905925];; for n in [11..30] do a[n]:=24*Sum([1..9], j-> a[n-j]) -300*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 26 2019

KEYWORD

nonn

AUTHOR

John Cannon and N. J. A. Sloane, Dec 03 2009

STATUS

approved

A166420

Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

+10
1

1, 26, 650, 16250, 406250, 10156250, 253906250, 6347656250, 158691406250, 3967285156250, 99182128906250, 2479553222655925, 61988830566390000, 1549720764159547200, 38743019103983610000, 968575477599463500000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170745, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (24,24,24,24,24,24,24,24,24,24,-300).

FORMULA

G.f.: (t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

From G. C. Greubel, Jan 17 2023: (Start)

a(n) = 24*Sum_{j=1..10} a(n-j) - 300*a(n-11).

G.f.: (1+x)*(1-x^11)/(1 - 25*x + 324*x^11 - 300*x^12). (End)

MATHEMATICA

With[{p=300, q=24}, CoefficientList[Series[(1+t)*(1-t^11)/(1-(q+1)*t + (p+q)*t^11-p*t^12), {t, 0, 40}], t]] (* G. C. Greubel, May 13 2016; Jul 25 2024 *)

coxG[{11, 300, -24}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Dec 31 2017 *)

PROG

(Magma)

R<x>:=PowerSeriesRing(Integers(), 30);

Coefficients(R!( (1+x)*(1-x^11)/(1-25*x+324*x^11-300*x^12) )); // G. C. Greubel, Jul 25 2024

(SageMath)

def A166420_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( (1+x)*(1-x^11)/(1-25*x+324*x^11-300*x^12) ).list()

A166420_list(30) # G. C. Greubel, Jul 25 2024

CROSSREFS

Cf. A154638, A169452, A170745.

KEYWORD

nonn

AUTHOR

John Cannon and N. J. A. Sloane, Dec 03 2009

STATUS

approved

A166613

Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

+10
1

1, 26, 650, 16250, 406250, 10156250, 253906250, 6347656250, 158691406250, 3967285156250, 99182128906250, 2479553222656250, 61988830566405925, 1549720764160140000, 38743019104003297200, 968575477600077360000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170745, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, -300).

FORMULA

G.f.: (t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^12 - 24*t^11 - 24*t^10 - 24*t^9 -24*t^8 -24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 -24*t + 1).

MATHEMATICA

CoefficientList[Series[(t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1), {t, 0, 50}], t] (* G. C. Greubel, May 19 2016 *)

coxG[{12, 300, -24}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 14 2018 *)

KEYWORD

nonn

AUTHOR

John Cannon and N. J. A. Sloane, Dec 03 2009

STATUS

approved

A167079

Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

+10
1

1, 26, 650, 16250, 406250, 10156250, 253906250, 6347656250, 158691406250, 3967285156250, 99182128906250, 2479553222656250, 61988830566406250, 1549720764160155925, 38743019104003890000, 968575477600097047200

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170745, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, -300).

FORMULA

G.f.: (t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^13 - 24*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

MATHEMATICA

CoefficientList[Series[(t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^13 - 24*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1), {t, 0, 50}], t] (* G. C. Greubel, May 31 2016 *)

KEYWORD

nonn

AUTHOR

John Cannon and N. J. A. Sloane, Dec 03 2009

STATUS

approved

A167225

Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^14 = I.

+10
1

1, 26, 650, 16250, 406250, 10156250, 253906250, 6347656250, 158691406250, 3967285156250, 99182128906250, 2479553222656250, 61988830566406250, 1549720764160156250, 38743019104003905925, 968575477600097640000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170745, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, -300).

FORMULA

G.f.: (t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^14 - 24*t^13 - 24*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

MATHEMATICA

CoefficientList[Series[(t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/ (300*t^14 - 24*t^13 - 24*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jun 05 2016 *)

coxG[{14, 300, -24}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Sep 16 2018 *)

KEYWORD

nonn

AUTHOR

John Cannon and N. J. A. Sloane, Dec 03 2009

STATUS

approved

A167697

Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^15 = I.

+10
1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170745, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, -300).

FORMULA

G.f.: (t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^15 - 24*t^14 - 24*t^13 - 24*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

MATHEMATICA

coxG[{15, 300, -24}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Aug 15 2015 *)

CoefficientList[Series[(t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(300*t^15 - 24*t^14 - 24*t^13 - 24*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jun 20 2016 *)

KEYWORD

nonn

AUTHOR

John Cannon and N. J. A. Sloane, Dec 03 2009

STATUS

approved

A167941

Number of reduced words of length n in Coxeter group on 26 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

+10
1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The initial terms coincide with those of A170745, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,-300).

FORMULA

G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/( 300*t^16 - 24*t^15 - 24*t^14 - 24*t^13 - 24*t^12 - 24*t^11 - 24*t^10 - 24*t^9 - 24*t^8 - 24*t^7 - 24*t^6 - 24*t^5 - 24*t^4 - 24*t^3 - 24*t^2 - 24*t + 1).

From G. C. Greubel, Sep 08 2023: (Start)

G.f.: (1+t)*(1-t^16)/(1 - 25*t + 324*t^16 - 300*t^17).

a(n) = 24*Sum_{j=1..15} a(n-j) - 300*a(n-16). (End)

MATHEMATICA

coxG[{16, 300, -24}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 01 2021 *)

CoefficientList[Series[(1+t)*(1-t^16)/(1-25*t+324*t^16-300*t^17), {t, 0, 50}], t] (* G. C. Greubel, Sep 08 2023 *)

PROG

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^16)/(1-25*x+324*x^16-300*x^17) )); // G. C. Greubel, Sep 08 2023

(SageMath)

def A167941_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( (1+x)*(1-x^16)/(1-25*x+324*x^16-300*x^17) ).list()

A167941_list(40) # G. C. Greubel, Sep 08 2023

CROSSREFS

Cf. A154638, A169452, A170758.

Cf. A167881, A167882, A167896 - A167900, A167908, A167914, A167916, A167919, A167922, A167923, A167924, A167926, A167927, A167929, A167931, A167933, A167935, A167937, A167938, A167940, A167942 - A167947, A167949 - A167962, A167978, A167980, A167988, A167989.

KEYWORD

nonn

AUTHOR

John Cannon and N. J. A. Sloane, Dec 03 2009

STATUS

approved

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