A179070 -id:A179070 - cp's OEIS frontend

A265071 Coordination sequence for (3,3,4) tiling of hyperbolic plane.

1, 3, 6, 10, 15, 22, 31, 44, 62, 87, 122, 171, 240, 336, 471, 660, 925, 1296, 1816, 2545, 3566, 4997, 7002, 9812, 13749, 19266, 26997, 37830, 53010, 74281, 104088, 145855, 204382, 286394, 401315, 562350, 788003, 1104204, 1547286, 2168163, 3038178, 4257303, 5965624, 8359440, 11713819, 16414204, 23000705, 32230160

Offset: 0

N. J. A. Sloane, Dec 29 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

I:=[1,3,6,10,15,22,31]; [n le 7 select I[n] else Self(n-2)+Self(n-3)+Self(n-4)- Self(n-6): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015

CoefficientList[Series[(x^3 + x^2 + x + 1) (x^2 + x + 1) (x + 1)/(x^6 - x^4 - x^3 - x^2 + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)

x='x+O('x^50); Vec((x^3+x^2+x+1)*(x^2+x+1)*(x+1)/(x^6-x^4-x^3-x^2+1)) \\ G. C. Greubel, Aug 07 2017

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

a(n) = a(n-2)+a(n-3)+a(n-4)-a(n-6) for n>6. - Vincenzo Librandi, Dec 30 2015

A265072 Coordination sequence for (3,3,5) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 6, 10, 16, 25, 38, 57, 86, 130, 196, 295, 444, 669, 1008, 1518, 2286, 3443, 5186, 7811, 11764, 17718, 26686, 40193, 60536, 91175, 137322, 206826, 311508, 469173, 706638, 1064293, 1602970, 2414290, 3636248, 5476683, 8248628, 12423553, 18711556, 28182142, 42446130, 63929631, 96286698, 145020831, 218421048

Offset: 0

N. J. A. Sloane, Dec 29 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,-1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

CoefficientList[Series[(x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1)/(x^6 - x^5 - x^3 - x + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)
LinearRecurrence[{1,0,1,0,1,-1},{1,3,6,10,16,25,38},50] (* Harvey P. Dale, Oct 07 2022 *)

x='x+O('x^50); Vec((x^2+x+1)*(x^4+x^3+x^2+x+1)/(x^6-x^5-x^3-x+1)) \\ G. C. Greubel, Aug 07 2017

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

A265073 Coordination sequence for (3,3,6) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 6, 10, 16, 26, 41, 64, 99, 154, 240, 374, 582, 905, 1408, 2191, 3410, 5306, 8256, 12846, 19989, 31104, 48399, 75310, 117184, 182342, 283730, 441493, 686976, 1068955, 1663326, 2588186, 4027296, 6266594, 9751009, 15172864, 23609435, 36736994, 57163872, 88948710, 138406878, 215365281, 335114880, 521448871

Offset: 0

N. J. A. Sloane, Dec 29 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 1, 1, -1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

CoefficientList[Series[(x^3 + 1) (x^2 + x + 1) (x + 1)/(x^6 - x^5 - x^4 + x^3 - x^2 - x + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)

x='x+O('x^50); Vec((x^3+1)*(x^2+x+1)*(x+1)/(x^6-x^5-x^4+x^3-x^2-x+1)) \\ G. C. Greubel, Aug 07 2017

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

A265074 Coordination sequence for (3,3,7) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 6, 10, 16, 26, 42, 67, 106, 167, 264, 418, 662, 1048, 1658, 2623, 4150, 6567, 10392, 16444, 26020, 41172, 65148, 103087, 163120, 258113, 408424, 646268, 1022620, 1618140, 2560460, 4051537, 6410938, 10144329, 16051850, 25399600, 40190986, 63596094, 100631100, 159233337, 251962422, 398692029, 630869210

Offset: 0

N. J. A. Sloane, Dec 29 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,0,1,-1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

I:=[1,3,6,10,16,26,42,67,106]; [n le 9 select I[n] else Self(n-1)+Self(n-3)+Self(n-5)+Self(n-7)-Self(n-8): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015

CoefficientList[Series[(x^2 + x + 1) (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/(x^8 - x^7 - x^5 - x^3 - x + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)

x='x+O('x^50); Vec((x^2+x+1)*(x^6+x^5+x^4+x^3+x^2+x+1)/(x^8-x^7-x^5-x^3-x+1)) \\ G. C. Greubel, Aug 07 2017

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

a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)-a(n-8) for n>8. - Vincenzo Librandi, Dec 30 2015

A265075 Coordination sequence for (3,4,4) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 6, 11, 18, 29, 46, 73, 116, 183, 290, 459, 726, 1149, 1818, 2877, 4552, 7203, 11398, 18035, 28538, 45157, 71454, 113065, 178908, 283095, 447954, 708819, 1121598, 1774757, 2808282, 4443677, 7031440, 11126179, 17605478, 27857979, 44080994, 69751437, 110370990, 174645225, 276349380, 437280663, 691929826

Offset: 0

N. J. A. Sloane, Dec 29 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
M. O'Keeffe, Coordination sequences for hyperbolic tilings, Zeitschrift für Kristallographie, 213 (1998), 135-140 (see last table, row 6.8.8H).
Index entries for linear recurrences with constant coefficients, signature (0, 1, 2, 1, 0, -1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

CoefficientList[Series[(x^3 + x^2 + x + 1) (x^2 + x + 1) (x + 1)/(x^6 - x^4 - 2 x^3 - x^2 + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)

x='x+O('x^50); Vec((x^3+x^2+x+1)*(x^2+x+1)*(x+1)/(x^6-x^4-2*x^3-x^2+1)) \\ G. C. Greubel, Aug 07 2017

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

A265076 Coordination sequence for (3,5,5) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 6, 11, 20, 35, 60, 103, 178, 307, 528, 909, 1566, 2697, 4644, 7997, 13772, 23717, 40842, 70333, 121120, 208579, 359190, 618555, 1065204, 1834371, 3158940, 5439959, 9368066, 16132595, 27781680, 47842381, 82388590, 141880057, 244329348, 420755613, 724576428, 1247781333, 2148784026, 3700386173, 6372375104

Offset: 0

N. J. A. Sloane, Dec 29 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 2, 0, 1, -1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

CoefficientList[Series[(x^2 + x + 1) (x^4 + x^3 + x^2 + x + 1)/(x^6 - x^5 - 2 x^3 - x + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)

Vec((x^2+x+1)*(x^4+x^3+x^2+x+1)/(x^6-x^5-2*x^3-x+1) + O(x^50)) \\ Michel Marcus, Dec 30 2015

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

A265077 Coordination sequence for (3,6,8) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 6, 11, 20, 37, 66, 117, 208, 371, 662, 1179, 2100, 3741, 6666, 11877, 21160, 37699, 67166, 119667, 213204, 379853, 676762, 1205749, 2148216, 3827355, 6818982, 12148995, 21645180, 38563997, 68707298, 122411917, 218094408, 388566507, 692287030, 1233408755, 2197494812, 3915152565, 6975406506, 12427688349

Offset: 0

N. J. A. Sloane, Dec 29 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,1,-1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

I:=[1,3,6,11,20,37,66]; [n le 7 select I[n] else Self(n-1)+Self(n-2)+Self(n-4) + Self(n-5)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015

CoefficientList[Series[(x^5 + x^4 + x^3 + x^2 + x + 1) (x + 1)/(x^6 - x^5 - x^4 - x^2 - x + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)

Vec((x^5+x^4+x^3+x^2+x+1)*(x+1)/(x^6-x^5-x^4-x^2-x+1) + O(x^50)) \\ Michel Marcus, Dec 30 2015

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

a(n) = a(n-1)+a(n-2)+a(n-4)+a(n-5)-a(n-6) for n>6. - Vincenzo Librandi, Dec 30 2015

A196456 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

Original entry on oeis.org

1, 3, 3, 4, 6, 4, 5, 14, 14, 5, 8, 30, 40, 30, 8, 12, 67, 131, 131, 67, 12, 17, 146, 383, 594, 383, 146, 17, 25, 320, 1232, 2930, 2930, 1232, 320, 25, 37, 706, 3966, 15274, 27320, 15274, 3966, 706, 37, 54, 1550, 12314, 76809, 249402, 249402, 76809, 12314, 1550, 54

Offset: 1

R. H. Hardin Oct 02 2011

Every 0 is next to 0 3's, every 1 is next to 1 0's, every 2 is next to 2 2's, every 3 is next to 3 4's, every 4 is next to 4 1's
Table starts
..1....3......4.......5..........8..........12............17..............25
..3....6.....14......30.........67.........146...........320.............706
..4...14.....40.....131........383........1232..........3966...........12314
..5...30....131.....594.......2930.......15274.........76809..........380093
..8...67....383....2930......27320......249402.......2032584........15952022
.12..146...1232...15274.....249402.....3707781......46887621.......578160718
.17..320...3966...76809....2032584....46887621.....926499600.....17962317556
.25..706..12314..380093...15952022...578160718...17962317556....539621963877
.37.1550..38745.1900745..128543371..7394096617..351103877665..16067939829696
.54.3403.122989.9527317.1060801262.96338727461.6945098996988.486076221745274

			Some solutions for n=6 k=4
..0..0..1..1....0..1..1..0....1..0..0..0....0..0..0..1....0..0..0..1
..1..1..1..0....2..2..1..0....1..0..0..0....1..1..1..1....0..0..2..2
..1..1..4..1....2..2..1..0....2..2..0..0....2..2..2..0....0..0..2..2
..0..0..1..1....0..1..1..0....2..2..0..0....2..0..2..0....2..2..1..0
..0..2..2..0....0..1..1..1....0..0..0..0....2..2..2..0....2..2..1..1
..1..2..2..1....1..1..0..1....1..1..1..1....0..0..0..0....0..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..161

Column 1 is A179070(n+2)

A196146 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,0,4,1,2 for x=0,1,2,3,4.

Original entry on oeis.org

1, 3, 3, 4, 5, 4, 5, 8, 8, 5, 8, 17, 7, 17, 8, 12, 34, 26, 26, 34, 12, 17, 62, 49, 83, 49, 62, 17, 25, 123, 85, 168, 168, 85, 123, 25, 37, 243, 178, 346, 481, 346, 178, 243, 37, 54, 471, 348, 865, 1031, 1031, 865, 348, 471, 54, 79, 924, 683, 2055, 2781, 2599, 2781, 2055

Offset: 1

R. H. Hardin Sep 28 2011

Every 0 is next to 0 3's, every 1 is next to 1 0's, every 2 is next to 2 4's, every 3 is next to 3 1's, every 4 is next to 4 2's
Table starts
..1...3....4.....5.....8.....12.....17......25.......37.......54........79
..3...5....8....17....34.....62....123.....243......471......924......1817
..4...8....7....26....49.....85....178.....348......683.....1349......2688
..5..17...26....83...168....346....865....2055.....4745....10866.....24967
..8..34...49...168...481...1031...2781....7666....19581....50767....133067
.12..62...85...346..1031...2599...7486...22292....61689...177057....510032
.17.123..178...865..2781...7486..24615...85533...266059...866323...2817802
.25.243..348..2055..7666..22292..85533..330889..1154587..4208560..15450981
.37.471..683..4745.19581..61689.266059.1154587..4431097.17980473..73718035
.54.924.1349.10866.50767.177057.866323.4208560.17980473.82611569.381145071

			Some solutions for n=6 k=4
.0.1.1.0...0.1.1.0...1.1.0.1...0.1.1.0...1.0.1.1
.1.3.1.0...0.1.1.0...0.1.1.1...0.1.1.0...1.1.1.0
.1.3.1.0...1.3.1.0...0.1.1.0...0.1.3.1...0.1.3.1
.0.1.1.0...1.3.1.0...0.1.1.0...0.1.3.1...0.1.3.1
.0.1.1.0...0.1.1.0...1.1.1.1...1.1.1.0...0.1.1.0
.1.3.1.0...1.3.1.0...1.0.0.1...1.0.1.1...1.3.1.0

R. H. Hardin, Table of n, a(n) for n = 1..312

Column 1 is A179070(n+2)

A196210 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 2,0,3,4,1 for x=0,1,2,3,4.

Original entry on oeis.org

1, 3, 3, 4, 5, 4, 5, 8, 8, 5, 8, 11, 7, 11, 8, 12, 18, 14, 14, 18, 12, 17, 28, 17, 35, 17, 28, 17, 25, 43, 27, 70, 70, 27, 43, 25, 37, 67, 42, 136, 135, 136, 42, 67, 37, 54, 105, 58, 305, 301, 301, 305, 58, 105, 54, 79, 164, 94, 611, 751, 899, 751, 611, 94, 164, 79, 116, 257, 137

Offset: 1

R. H. Hardin Sep 29 2011

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 3's, every 3 is next to 3 4's, every 4 is next to 4 1's
Table starts
..1...3...4....5.....8....12.....17......25.......37........54........79
..3...5...8...11....18....28.....43......67......105.......164.......257
..4...8...7...14....17....27.....42......58.......94.......137.......208
..5..11..14...35....70...136....305.....611.....1321......2832......6041
..8..18..17...70...135...301....751....1774.....4243.....10211.....24823
.12..28..27..136...301...899...2922....7680....24326.....71539....213378
.17..43..42..305...751..2922..10986...35529...128823....472599...1687338
.25..67..58..611..1774..7680..35529..136967...627285...2695504..11805281
.37.105..94.1321..4243.24326.128823..627285..3421139..17683925..93767603
.54.164.137.2832.10211.71539.472599.2695504.17683925.111262773.726204291

			Some solutions for n=6 k=4
..0..0..1..1....1..0..0..1....0..1..1..0....1..0..1..1....0..1..1..0
..1..1..1..0....1..0..0..1....0..1..1..0....1..1..1..0....1..1..1..0
..1..4..1..0....1..0..0..1....1..1..4..1....0..1..1..0....1..0..1..1
..0..1..1..0....1..0..0..1....1..0..1..1....0..1..1..0....1..0..1..1
..1..1..1..1....1..0..0..1....1..1..1..0....1..1..1..1....1..1..1..0
..1..0..0..1....1..0..0..1....0..1..1..0....1..0..0..1....0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..337

Column 1 is A179070(n+2)

cp's OEIS Frontend

A265071 Coordination sequence for (3,3,4) tiling of hyperbolic plane.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A265072 Coordination sequence for (3,3,5) tiling of hyperbolic plane.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A265073 Coordination sequence for (3,3,6) tiling of hyperbolic plane.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A265074 Coordination sequence for (3,3,7) tiling of hyperbolic plane.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A265075 Coordination sequence for (3,4,4) tiling of hyperbolic plane.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A265076 Coordination sequence for (3,5,5) tiling of hyperbolic plane.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A265077 Coordination sequence for (3,6,8) tiling of hyperbolic plane.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A196456 T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,0,2,4,1 for x=0,1,2,3,4.

Views

Author

Keywords

Comments

Examples