A003953 -id:A003953 - cp's OEIS frontend

A101229 Perfect inverse "3x+1 conjecture" (See comments for rules).

1, 2, 4, 1, 2, 4, 8, 16, 5, 10, 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

Offset: 1

Alexandre Wajnberg, Jan 22 2005

Perfect inverse "3x+1 conjecture": rule 1: multiply n by 2 to give n' = 2n. rule 2: when n'=(3x+1), do n"= (n'-1)/3 (n" integer) Additional rule: rule 2 is applied once for any number n' (otherwise, the sequence beginning with 1 would be the cycle "1 2 4 1 2 4 1 2 4 1..."); then apply rule 1.
This gives a particular sequence of hailstone numbers which may be considered as a central axis for all the hailstone number sequences. The perfect inverse "3x+1 conjecture" falls rapidly into the sequence 3 6 12 24 48 96... which will never give a number to which apply the 2nd rule.
a(n) for n >= 11 written in base 2: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-11) times 0 (see A003953(n-10)). - Jaroslav Krizek, Aug 17 2009

			The first 4 is followed by 1 because 4 = 3*1 + 1, so rule 2: (4-1)/3 = 1;
the second 4 is followed by 8 because the 2nd rule has already been applied, so rule 1: 4*2 = 8.

R. K. Guy, Collatz's Sequence, Section E16 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 215-218, 1994.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Collatz Problem
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A070165, A006577, A006667, A006666, A070167.

R:=PowerSeriesRing(Integers(), 45); Coefficients(R!( x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1) )); // G. C. Greubel, Mar 20 2019

Rest[CoefficientList[Series[x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1), {x, 0, 45}], x]] (* G. C. Greubel, Mar 20 2019 *)
LinearRecurrence[{2},{1,2,4,1,2,4,8,16,5,10,3},40] (* Harvey P. Dale, May 06 2023 *)

my(x='x+O('x^45)); Vec(x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1)) \\ G. C. Greubel, Mar 20 2019

a=(x*(17*x^10+27*x^8+7*x^3-1)/(2*x-1)).series(x, 45).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Mar 20 2019

a(n) = 3*2^(n-11) = 2^(n-11) + 2^(n-10) for n >= 11. - Jaroslav Krizek, Aug 17 2009
From Colin Barker, Apr 28 2013: (Start)
a(n) = 2*a(n-1) for n>11.
G.f.: x*(17*x^10+27*x^8+7*x^3-1) / (2*x-1). (End)

More terms from Joshua Zucker, May 18 2006

Edited by G. C. Greubel, Mar 20 2019

A158497 Triangle T(n,k) formed by the coordination sequences and the number of leaves for trees.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 12, 1, 4, 12, 36, 108, 1, 5, 20, 80, 320, 1280, 1, 6, 30, 150, 750, 3750, 18750, 1, 7, 42, 252, 1512, 9072, 54432, 326592, 1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, 430467210, 3874204890

Offset: 0

Thomas Wieder, Mar 20 2009

Consider the k-fold Cartesian products CP(n,k) of the vector A(n) = [1, 2, 3, ..., n].
An element of CP(n,k) is a n-tuple T_t of the form T_t = [i_1, i_2, i_3, ..., i_k] with t=1, .., n^k.
We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.
For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).
The test on i_j > i_(j+1) generates A158498. One gets the Pascal triangle A007318 if the indicator function tests whether for any two i_j, i_(j+1) in T_t one has i_j >= i_(j+1).
Use of other indicator functions can also calculate the Bell numbers A000110, A000045 or A000108.

			Array, A(n, k) = n*(n-1)^(k-1) for n > 1, A(n, k) = 1 otherwise, begins as:
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  1,   1,    1,     1,      1,       1,        1,        1, ... A000012;
  1,  2,   2,    2,     2,      2,       2,        2,        2, ... A040000;
  1,  3,   6,   12,    24,     48,      96,      192,      384, ... A003945;
  1,  4,  12,   36,   108,    324,     972,     2916,     8748, ... A003946;
  1,  5,  20,   80,   320,   1280,    5120,    20480,    81920, ... A003947;
  1,  6,  30,  150,   750,   3750,   18750,    93750,   468750, ... A003948;
  1,  7,  42,  252,  1512,   9072,   54432,   326592,  1959552, ... A003949;
  1,  8,  56,  392,  2744,  19208,  134456,   941192,  6588344, ... A003950;
  1,  9,  72,  576,  4608,  36864,  294912,  2359296, 18874368, ... A003951;
  1, 10,  90,  810,  7290,  65610,  590490,  5314410, 47829690, ... A003952;
  1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, ............. A003953;
  1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, ............. A003954;
  1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, ............. A170732;
  ... ;
The triangle begins as:
  1
  1, 1;
  1, 2,  2;
  1, 3,  6,  12;
  1, 4, 12,  36,  108;
  1, 5, 20,  80,  320,  1280;
  1, 6, 30, 150,  750,  3750,  18750;
  1, 7, 42, 252, 1512,  9072,  54432, 326592;
  1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344;
  ...;
T(3,3) = 12 counts the triples (1,2,1), (1,2,3), (1,3,1), (1,3,2), (2,1,2), (2,1,3), (2,3,1), (2,3,2), (3,1,2), (3,1,3), (3,2,1), (3,2,3) out of a total of 3^3 = 27 triples in the CP(3,3).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Array rows n: A170733 (n=14), ..., A170769 (n=50).
Columns k: A000012(n) (k=0), A000027(n) (k=1), A002378(n-1) (k=2), A011379(n-1) (k=3), A179824(n) (k=4), A101362(n-1) (k=5), 2*A168351(n-1) (k=6), 2*A168526(n-1) (k=7), 2*A168635(n-1) (k=8), 2*A168675(n-1) (k=9), 2*A170783(n-1) (k=10), 2*A170793(n-1) (k=11).
Diagonals k: A055897 (k=n), A055541 (k=n-1), A373395 (k=n-2), A379612 (k=n-3).
Sums: (-1)^n*A065440(n) (signed row).
Cf. A000045, A000108, A000110, A007318, A079901, A158498.

A158497:= func< n,k | k le 1 select n^k else n*(n-1)^(k-1) >;
[A158497(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2025

A158497[n_, k_]:= If[n<2 || k==0, 1, n*(n-1)^(k-1)];
Table[A158497[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 18 2025 *)

def A158497(n,k): return n^k if k<2 else n*(n-1)^(k-1)
print(flatten([[A158497(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 18 2025

T(n, k) = (n-1)^(k-1) + (n-1)^k = n*A079901(n-1,k-1), k > 0.

Sum_{k=0..n} T(n,k) = (n*(n-1)^n - 2)/(n-2), n > 2.

Edited by R. J. Mathar, Mar 31 2009

More terms added by G. C. Greubel, Mar 18 2025

A162987 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 11, 110, 1100, 10945, 108900, 1083555, 10781100, 107269470, 1067306625, 10619454780, 105661128375, 1051303881870, 10460231387100, 104076892111005, 1035541095642900, 10303395297584895, 102516409155629700, 1020014649794722230, 10148910738927500925

Offset: 0

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

The initial terms coincide with those of A003953, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (9,9,9,-45).

a:=[11,110,1100,10945];; for n in [5..20] do a[n]:=9*(a[n-1]+a[n-2] +a[n-3] -5*a[n-4]); od; Concatenation([1], a); # G. C. Greubel, Apr 28 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^4)/(1-10*x+54*x^4-45*x^5) )); // G. C. Greubel, Apr 28 2019

CoefficientList[Series[(1+x)*(1-x^4)/(1-10*x+54*x^4-45*x^5), {x,0,20}], x] (* or *) coxG[{4, 45, -9}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 28 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^4)/(1-10*x+54*x^4-45*x^5)) \\ G. C. Greubel, Apr 28 2019

((1+x)*(1-x^4)/(1-10*x+54*x^4-45*x^5)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

G.f.: (t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(45*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).
From G. C. Greubel, Apr 28 2019: (Start)
a(n) = 9*(a(n-1) + a(n-2) + a(n-3) - 5*a(n-4)).
G.f.: (1+x)*(1-x^4)/(1 - 10*x + 54*x^4 - 45*x^5). (End)

A162760 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I.

Original entry on oeis.org

1, 11, 110, 1045, 9900, 93555, 884070, 8353125, 78924780, 745717995, 7045894350, 66572896005, 629011803420, 5943197049075, 56154099352230, 530570136457845, 5013074255082300, 47365865053010955, 447534797632236270

Offset: 0

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

nonn

The initial terms coincide with those of A003953, although the two sequences are eventually different.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9, 9, -45).

I:=[1,11,110,1045]; [n le 4 select I[n] else 9*Self(n-1) +9*Self(n-2)-45*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 01 2017

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+x)*(1-x^3)/(1-10*x+54*x^3-45*x^4) )); // G. C. Greubel, Apr 26 2019

Join[{1}, LinearRecurrence[{9, 9, -45}, {11, 110, 1045}, 19]] (* Vincenzo Librandi, Apr 01 2017 *)
CoefficientList[Series[(1+x)*(1-x^3)/(1-10*x+54*x^3-45*x^4), {x,0,20}],x] (* or *) coxG[{3, 45, -9}] (* The coxG program is at A169452 *) (* G. C. Greubel, Apr 26 2019 *)

my(x='x+O('x^20)); Vec((1+x)*(1-x^3)/(1-10*x+54*x^3-45*x^4)) \\ G. C. Greubel, Apr 26 2019

((1+x)*(1-x^3)/(1-10*x+54*x^3-45*x^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Apr 26 2019

G.f.: (t^3 + 2*t^2 + 2*t + 1)/(45*t^3 - 9*t^2 - 9*t + 1).

G.f.: (1+x)*(1-x^3)/(1 - 10*x + 54*x^3 - 45*x^4). - G. C. Greubel, Apr 26 2019

A163404 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

Original entry on oeis.org

1, 11, 110, 1100, 11000, 109945, 1098900, 10983555, 109781100, 1097266500, 10967222970, 109617836625, 1095634704780, 10950913128375, 109454819042250, 1094005337374620, 10934627535602100, 109292043884611005

Offset: 0

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

The initial terms coincide with those of A003953, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (9,9,9,9,-45).

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-10*x+54*x^5-45*x^6) )); // G. C. Greubel, May 12 2019

CoefficientList[Series[(1 + x)*(1-x^5)/(1-10*x+54*x^5-45*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{9, 9, 9, 9, -45}, {1, 11, 110, 1100, 11000, 109945}, 30] (* G. C. Greubel, Dec 21 2016 *)
coxG[{5, 45, -9}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-10*x+54*x^5-45*x^6)) \\ G. C. Greubel, Dec 21 2016

((1+x)*(1-x^5)/(1-10*x+54*x^5-45*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(45*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).

a(n) = 9*a(n-1)+9*a(n-2)+9*a(n-3)+9*a(n-4)-45*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163955 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 11, 110, 1100, 11000, 110000, 1099945, 10998900, 109983555, 1099781100, 10997266500, 109967220000, 1099617752970, 10995633086625, 109950886704780, 1099454428128375, 10993999919042250, 109934555837535000

Offset: 0

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

nonn

The initial terms coincide with those of A003953, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (9,9,9,9,9,-45).

a:=[11,110,1100,11000,110000,1099945];; for n in [7..30] do a[n]:=9*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -45*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7) )); // G. C. Greubel, Aug 10 2019

seq(coeff(series((1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

CoefficientList[Series[(1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 13 2017 *)
coxG[{6, 45, -9}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 10 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7)) \\ G. C. Greubel, Aug 13 2017

def A163955_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-10*t+54*t^6-45*t^7)).list()
A163955_list(30) # G. C. Greubel, Aug 10 2019

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(45*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).

a(n) = -45*a(n-6) + 9*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021

A164590 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

Original entry on oeis.org

1, 11, 110, 1100, 11000, 110000, 1100000, 10999945, 109998900, 1099983555, 10999781100, 109997266500, 1099967220000, 10999617750000, 109995633002970, 1099950885086625, 10999454401704780, 109993999528128375

Offset: 0

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

nonn

The initial terms coincide with those of A003953, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (9,9,9,9,9,9,-45).

a:=[11, 110, 1100, 11000, 110000, 1100000, 10999945];; for n in [8..30] do a[n]:=9*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -45*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-10*t+54*t^7-45*t^8) )); // G. C. Greubel, Aug 28 2019

seq(coeff(series((1+t)*(1-t^7)/(1-10*t+54*t^7-45*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019

CoefficientList[Series[(1+t)*(1-t^7)/(1-10*t+54*t^7-45*t^8), {t, 0, 30}], t] (* G. C. Greubel, Aug 12 2017 *)
coxG[{7, 45, -9}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 28 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-10*t+54*t^7-45*t^8)) \\ G. C. Greubel, Aug 12 2017

def A164590_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-10*t+54*t^7-45*t^8)).list()
A164590_list(30) # G. C. Greubel, Aug 28 2019

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(45*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).

a(n) = -45*a(n-7) + 9*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021

A165796 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

Original entry on oeis.org

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 10999999945, 109999998900, 1099999983555, 10999999781100, 109999997266500, 1099999967220000, 10999999617750000, 109999995633000000

Offset: 0

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

nonn

The initial terms coincide with those of A003953, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
a(0) = a(1) = 1 (mod 5), a(n) = 0 (mod 5) for n>=2. - G. C. Greubel, Apr 08 2016

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (9,9,9,9,9,9,9,9,9,-45).

a:=[11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 10999999945];; for n in [11..20] do a[n]:=9*Sum([1..9], j-> a[n-j]) -45*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 22 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-10*t+54*t^10-45*t^11) )); // G. C. Greubel, Sep 22 2019

seq(coeff(series((1+t)*(1-t^10)/(1-10*t+54*t^10-45*t^11), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Sep 22 2019

CoefficientList[Series[(1+t)*(1-t^10)/(1-10*t+54*t^10-45*t^11), {t, 0, 25}], t] (* G. C. Greubel, Apr 08 2016 *)
coxG[{10, 45, -9}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 22 2019 *)

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-10*t+54*t^10-45*t^11)) \\ G. C. Greubel, Sep 22 2019

def A165796_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^10)/(1-10*t+54*t^10-45*t^11)).list()
A165796_list(20) # G. C. Greubel, Sep 22 2019

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)/(45*t^10 - 9*t^9 - 9*t^8 - 9*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).

A166369 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

Original entry on oeis.org

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 109999999945, 1099999998900, 10999999983555, 109999999781100, 1099999997266500, 10999999967220000, 109999999617750000

Offset: 0

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

nonn

The initial terms coincide with those of A003953, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (9,9,9,9,9,9,9,9,9,9,-45).

Cf. A003953, A154638, A169452.

R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (1+x)*(1-x^11)/(1-10*x+54*x^11-45*x^12) )); // G. C. Greubel, Jul 23 2024

With[{p=45, q=9}, 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 10 2016; Jul 23 2024 *)
coxG[{11,45,-9}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jun 12 2016 *)

def A166369_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^11)/(1-10*x+54*x^11-45*x^12) ).list()
A166369_list(30) # G. C. Greubel, Jul 23 2024

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)/(45*t^11 - 9*t^10 - 9*t^9 - 9*t^8 - 9*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).
From G. C. Greubel, Jul 23 2024: (Start)
a(n) = 9*Sum_{j=1..10} a(n-j) - 45*a(n-11).
G.f.: (1+t)*(1-t^11)/(1 - 10*t + 54*t^11 - 45*t^12). (End)

A166551 Number of reduced words of length n in Coxeter group on 11 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

Original entry on oeis.org

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 110000000000, 1099999999945, 10999999998900, 109999999983555, 1099999999781100, 10999999997266500, 109999999967220000

Offset: 0

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

nonn

The initial terms coincide with those of A003953, although the two sequences are eventually different.

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

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (9,9,9,9,9,9,9,9,9,9,9,-45).

Cf. A003953, A154638, A169452.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-10*x+54*x^12-45*x^13) )); // G. C. Greubel, Aug 23 2024

CoefficientList[Series[(1+t)*(1-t^12)/(1-10*t+54*t^12-45*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 17 2016; Aug 23 2024 *)
coxG[{12,45,-9}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 20 2020 *)

def A166551_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-10*x+54*x^12-45*x^13) ).list()
A166551_list(30) # G. C. Greubel, Aug 23 2024

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)/(45*t^12 - 9*t^11 - 9*t^10 - 9*t^9 - 9*t^8 - 9*t^7 - 9*t^6 - 9*t^5 - 9*t^4 - 9*t^3 - 9*t^2 - 9*t + 1).
From G. C. Greubel, Aug 23 2024: (Start)
a(n) = 9*Sum_{j=1..11} a(n-j) - 45*a(n-12).
G.f.: (1+x)*(1-x^12)/(1 - 10*x + 54*x^12 - 45*x^13). (End)

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