A003947 -id:A003947 - cp's OEIS frontend

A170686 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^50 = I.

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555520, 87960930222080

Offset: 0

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

nonn

The initial terms coincide with those of A003947, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.
About the initial comment, first disagreement is at index 50 and the difference is 10.  - Vincenzo Librandi, Dec 09 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, -6).

With[{num = Total[2 t^Range[49]] + t^50 + 1, den = Total[-3 t^Range[49]] + 6 t^50 + 1}, CoefficientList[Series[num/den, {t, 0, 40}], t]] (* Vincenzo Librandi, Dec 09 2012 *)

G.f. (t^50 + 2*t^49 + 2*t^48 + 2*t^47 + 2*t^46 + 2*t^45 + 2*t^44 + 2*t^43 +
2*t^42 + 2*t^41 + 2*t^40 + 2*t^39 + 2*t^38 + 2*t^37 + 2*t^36 + 2*t^35 +
2*t^34 + 2*t^33 + 2*t^32 + 2*t^31 + 2*t^30 + 2*t^29 + 2*t^28 + 2*t^27 +
2*t^26 + 2*t^25 + 2*t^24 + 2*t^23 + 2*t^22 + 2*t^21 + 2*t^20 + 2*t^19 +
2*t^18 + 2*t^17 + 2*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)/(6*t^50 - 3*t^49 - 3*t^48 - 3*t^47 - 3*t^46 - 3*t^45 - 3*t^44 -
3*t^43 - 3*t^42 - 3*t^41 - 3*t^40 - 3*t^39 - 3*t^38 - 3*t^37 - 3*t^36 -
3*t^35 - 3*t^34 - 3*t^33 - 3*t^32 - 3*t^31 - 3*t^30 - 3*t^29 - 3*t^28 -
3*t^27 - 3*t^26 - 3*t^25 - 3*t^24 - 3*t^23 - 3*t^22 - 3*t^21 - 3*t^20 -
3*t^19 - 3*t^18 - 3*t^17 - 3*t^16 - 3*t^15 - 3*t^14 - 3*t^13 - 3*t^12 -
3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3
- 3*t^2 - 3*t + 1).

A269696 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 6", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280

Offset: 0

Robert Price, Mar 03 2016

Initialized with a single black (ON) cell at stage zero.
Rules 38, 70, 102, 134, 166, 198 and 230 also generate this sequence.
Apparently a duplicate of A003947. - R. J. Mathar, Mar 09 2016

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A269695.

rule=6; stages=300;
ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

Conjectures from Colin Barker, Mar 08 2016: (Start)
a(n) = 5*4^(n-1) for n>0.
a(n) = 4*a(n-1) for n>1.
G.f.: (1+x) / (1-4*x).
(End)

a(9)-a(15) from Lars Blomberg, Apr 12 2016

A277451 Number of edges in geodesic dome generated from icosahedron by recursively dividing each triangle in 4.

Original entry on oeis.org

1, 30, 120, 480, 1920, 7680, 30720, 122880, 491520, 1966080, 7864320, 31457280, 125829120, 503316480, 2013265920, 8053063680, 32212254720, 128849018880, 515396075520, 2061584302080, 8246337208320, 32985348833280, 131941395333120, 527765581332480

Offset: 0

Jonah Caplan, Oct 16 2016

The new triangles are generated by placing new vertices at the midpoints of each edge in the old triangle.

			n = 1 is the icosahedron with 30 sides. After dividing each face in 4, there are 120 sides in the next iteration.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4).

A122973 is the number of vertices, A003947 is the number of faces starting from 20.

{1}~Join~NestList[4 # &, 30, 22] (* or *)
CoefficientList[Series[(1 + 26 x)/(1 - 4 x), {x, 0, 23}], x] (* Michael De Vlieger, Oct 21 2016 *)

Vec((1+26*x)/(1-4*x) + O(x^30)) \\ Colin Barker, Oct 20 2016

a = [1] + [30 * 4 ** (n-1) for n in range(1,24)]

a(n) = 1 if n=0, else 30*4^(n-1).
From Colin Barker, Oct 20 2016: (Start)
a(n) = 4*a(n-1) for n>1.
G.f.: (1+26*x) / (1-4*x). (End)

A287811 Number of septenary sequences of length n such that no two consecutive terms have distance 5.

Original entry on oeis.org

1, 7, 45, 291, 1881, 12159, 78597, 508059, 3284145, 21229047, 137226717, 887047443, 5733964809, 37064931183, 239591481525, 1548743682699, 10011236540769, 64713650292711, 418315611378573, 2704034619149571, 17479154549033145, 112987031151647583

Offset: 0

David Nacin, Jun 01 2017

			For n=2 the a(2) = 49-4 = 45 sequences contain every combination except these four: 05, 50, 16, 61.

Index entries for linear recurrences with constant coefficients, signature (6, 3).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473.

Cf. A287804-A287819.

Mathematica
```
LinearRecurrence[{6, 3}, {1,7}, 40]
```

def a(n):
 if n in [0, 1]:
  return [1, 7][n]
 return 6*a(n-1)-3*a(n-2)

a(n) = 6*a(n-1) + 3*a(n-2), a(0)=1, a(1)=7.
G.f.: (1 + x)/(1 - 6*x - 3*x^2).
a(n) = A090018(n-1)+A090018(n). - R. J. Mathar, Oct 20 2019

A287838 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 8.

Original entry on oeis.org

1, 11, 115, 1205, 12625, 132275, 1385875, 14520125, 152130625, 1593906875, 16699721875, 174966753125, 1833166140625, 19206495171875, 201230782421875, 2108340300078125, 22089556912890625, 231437270629296875, 2424820490857421875, 25405391261720703125

Offset: 0

David Nacin, Jun 07 2017

In general, the number of sequences on {0,1,...,10} such that no two consecutive terms have distance 6+k for k in {0,1,2,3,4} has generating function (-1 - x)/(-1 + 10*x + (2*k+1)*x^2).

Colin Barker, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (10,5).

Cf. A040000, A003945, A083318, A078057, A003946, A126358, A003946, A055099, A003947, A015448, A126473. A287804-A287819. A287825-A287839.

LinearRecurrence[{10, 5}, {1, 11, 115}, 20]

Vec((1 + x) / (1 - 10*x - 5*x^2) + O(x^40)) \\ Colin Barker, Nov 25 2017

def a(n):
 if n in [0,1,2]:
  return [1, 11, 115][n]
 return 10*a(n-1) + 5*a(n-2)

For n > 2, a(n) = 10*a(n-1) + 5*a(n-2), a(0)=1, a(1)=11, a(2)=115.
G.f.: (-1 - x)/(-1 + 10*x + 5*x^2).
a(n) = (((5-sqrt(30))^n*(-6+sqrt(30)) + (5+sqrt(30))^n*(6+sqrt(30)))) / (2*sqrt(30)). - Colin Barker, Nov 25 2017

A162925 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

Original entry on oeis.org

1, 5, 20, 80, 310, 1200, 4650, 18000, 69690, 269820, 1044630, 4044420, 15658470, 60623640, 234711810, 908715240, 3518201250, 13621143060, 52735907790, 204173464860, 790482339630, 3060448278480, 11848896802170, 45874441471680

Offset: 0

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

nonn

The initial terms coincide with those of A003947, although the two sequences are eventually different.

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

Index entries for linear recurrences with constant coefficients, signature (3, 3, 3, -6).

coxG[{4,6,-3,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 28 2023 *)

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

A164706 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^8 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81910, 327600, 1310250, 5240400, 20959200, 83827200, 335270400, 1340928000, 5363097690, 21449933820, 85789908630, 343120332420, 1372324139280, 5488667867520, 21952156999680, 87798571269120

Offset: 0

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

nonn

The initial terms coincide with those of A003947, although the two sequences are eventually different.

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

Index entries for linear recurrences with constant coefficients, signature (3, 3, 3, 3, 3, 3, 3, -6).

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

A165185 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327670, 1310640, 5242410, 20969040, 83873760, 335485440, 1341903360, 5367459840, 21469224960, 85874442330, 343487939580, 1373912440470, 5495492494980, 21981340930320, 87922847594880

Offset: 0

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

nonn

The initial terms coincide with those of A003947, although the two sequences are eventually different.

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

Index entries for linear recurrences with constant coefficients, signature (3, 3, 3, 3, 3, 3, 3, 3, -6).

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

A168874 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^21 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138870, 21990232555440, 87960930221610

Offset: 0

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

nonn

The initial terms coincide with those of A003947, although the two sequences are eventually different.
First disagreement at index 21: a(21) = 5497558138870, A003947(21) = 5497558138880. - Klaus Brockhaus, Apr 04 2011
Computed with MAGMA using commands similar to those used to compute A154638.

Index entries for linear recurrences with constant coefficients, signature (3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, -6).

Cf. A003947 (G.f.: (1+x)/(1-4*x)).

coxG[{21,6,-3,30}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Mar 25 2015 *)

G.f.: (t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*t^17 + 2*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)/(6*t^21 - 3*t^20 - 3*t^19 - 3*t^18 - 3*t^17 - 3*t^16 - 3*t^15 - 3*t^14 - 3*t^13 - 3*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

A168922 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^22 = I.

Original entry on oeis.org

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280, 5368709120, 21474836480, 85899345920, 343597383680, 1374389534720, 5497558138880, 21990232555510, 87960930222000

Offset: 0

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

nonn

The initial terms coincide with those of A003947, although the two sequences are eventually different.
First disagreement at index 22: a(22) = 21990232555510, A003947(22) = 21990232555520. - Klaus Brockhaus, Apr 09 2011
Computed with MAGMA using commands similar to those used to compute A154638.

Index entries for linear recurrences with constant coefficients, signature (3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, -6).

Cf. A003947 (G.f.: (1+x)/(1-4*x)).

G.f.: (t^22 + 2*t^21 + 2*t^20 + 2*t^19 + 2*t^18 + 2*t^17 + 2*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)/(6*t^22 - 3*t^21 - 3*t^20 - 3*t^19 - 3*t^18 - 3*t^17 - 3*t^16 - 3*t^15 - 3*t^14 - 3*t^13 - 3*t^12 - 3*t^11 - 3*t^10 - 3*t^9 - 3*t^8 - 3*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 3*t^3 - 3*t^2 - 3*t + 1).

cp's OEIS Frontend

A170686 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^50 = I.

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A269696 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 6", based on the 5-celled von Neumann neighborhood.

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A277451 Number of edges in geodesic dome generated from icosahedron by recursively dividing each triangle in 4.

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A287811 Number of septenary sequences of length n such that no two consecutive terms have distance 5.

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A287838 Number of words of length n over the alphabet {0,1,...,10} such that no two consecutive terms have distance 8.

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A162925 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^4 = I.

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A164706 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^8 = I.

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A165185 Number of reduced words of length n in Coxeter group on 5 generators S_i with relations (S_i)^2 = (S_i S_j)^9 = I.

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