A096231 -id:A096231 - cp's OEIS frontend

A265069 Coordination sequence for (2,6,6) tiling of hyperbolic plane.

1, 3, 5, 8, 13, 21, 32, 47, 71, 108, 163, 245, 368, 555, 837, 1260, 1897, 2857, 4304, 6483, 9763, 14704, 22147, 33357, 50240, 75667, 113965, 171648, 258525, 389373, 586448, 883271, 1330327, 2003652, 3017771, 4545173, 6845648, 10310475, 15528973, 23388740, 35226617, 53056065, 79909632, 120354747, 181270579, 273018088

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^5 + x^4 + x^3 + x^2 + x + 1) (x + 1)/(x^6 - x^5 - x^3 - 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^3-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^3-x+1).

A265070 Coordination sequence for (2,6,infinity) tiling of hyperbolic plane.

Original entry on oeis.org

1, 3, 5, 8, 13, 21, 33, 51, 80, 126, 198, 311, 488, 766, 1203, 1889, 2966, 4657, 7312, 11481, 18027, 28305, 44443, 69782, 109568, 172038, 270125, 424136, 665956, 1045649, 1641823, 2577904, 4047689, 6355468, 9979021, 15668533, 24601905, 38628615, 60652616, 95233542, 149530690, 234785211, 368647368, 578830674

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).

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:=[1,3,5,8,13,21,33]; [n le 7 select I[n] else Self(n-1)+Self(n-3)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015

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

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

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

a(n) = a(n-1)+a(n-3)+a(n-5) for n>6. - Vincenzo Librandi, Dec 30 2015

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

Original entry on oeis.org

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

A134816 Padovan's spiral numbers.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426, 396655

Offset: 1

Omar E. Pol, Nov 13 2007

a(n) is the length of the edge of the n-th equilateral triangle in the Padovan triangle spiral.
Partial sums of A000931. - Juri-Stepan Gerasimov, Jul 17 2009
Rising diagonal sums of triangle A152198. - John Molokach, Jul 09 2013
a(n) is the number of pairs of rabbits living at month n with the following rules: a pair of rabbits born in month n begins to procreate in month n + 2, procreates again in month n + 3, and dies at the end of this month (each pair therefore gives birth to 2 pairs); the first pair is born in month 1. - Robert FERREOL, Oct 16 2017

			a(6)=3 because 6+4=10 and A000931(10)=3.
G.f. = x + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + ... - _Michael Somos_, Jan 01 2019

Muniru A Asiru, Table of n, a(n) for n = 1..1500
J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol. 17, No. 3 (2016).
Eric Fernando Bravo, On concatenations of Padovan and Perrin numbers, Math. Commun. (2023) Vol 28, 105-119.
Alain Faisant, On the Padovan sequence, arXiv:1905.07702 [math.NT], 2019.
Ed Harris, Pete McPartlan and Brady Haran, The Plastic Ratio, Numberphile video (2019).
Mariana Nagy, Simon R. Cowell, and Valeriu Beiu, On the Construction of 3D Fibonacci Spirals, Mathematics (2024) Vol. 12, No. 2, 201.
Christian Richter, Tilings of convex polygons by equilateral triangles of many different sizes, Discrete Mathematics 343.3 (2020): 111745. (See Section 2.1.)
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
S. J. Tedford, Combinatorial identities for the Padovan numbers, Fib. Q., 57:4 (2019), 291-298.
Wikipedia, Padovan triangles
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Cf. A060006.

a:=[1,1,1];; for n in [4..50] do a[n]:=a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Aug 12 2018

a:=proc(n, p, q) option remember:
if n<=p then 1
elif n<=q then a(n-1, p, q)+a(n-p, p, q)
else add(a(n-k, p, q), k=p..q) fi end:
seq(a(n, 2, 3), n=0..100); # Robert FERREOL, Oct 16 2017

Drop[ CoefficientList[ Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, 52}], x], 5] (* Robert G. Wilson v, Sep 30 2009 *)
a[n_]=Round[Root[23#^3-5#-1&,1]Root[#^3-#-1&,1]^n ];a[Range[100]] (* OR *)
LinearRecurrence[{0, 1, 1}, {1, 1, 1}, 100] (* Federico Provvedi, Feb 12 2025 *)

{a(n) = if( n>=0, polcoeff( (x + x^2) / (1 - x^2 - x^3) + x * O(x^n), n), polcoeff( (x + x^2) / (1 + x - x^3) + x * O(x^-n), -n))}; /* Michael Somos, Jan 01 2019 */

my(x='x+O('x^50)); Vec(x*(1+x)/(1-x^2-x^3)) \\ Joerg Arndt, Feb 07 2025

a(n) = A000931(n+4).
G.f.: x * (1 + x) / (1 - x^2 - x^3) = x / (1 - x / (1 - x^2 / (1 + x / (1 - x / (1 + x))))). - Michael Somos, Jan 03 2013
a(1)=a(2)=a(3)=1, a(n) = a(n-2) + a(n-3) for n > 3. - Robert FERREOL, Oct 16 2017
a(n) = round(x*rho^n), where the Silver constant rho = Limit_{n->oo} a(n+1)/a(n) = A060006, and x is the real solution of the cubic 23*x^3-5*x-1 = 0. - Federico Provvedi, Feb 12 2025

More terms from Robert G. Wilson v, Sep 30 2009

First comment clarified by Omar E. Pol, Aug 12 2018

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A265069 Coordination sequence for (2,6,6) tiling of hyperbolic plane.

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A265070 Coordination sequence for (2,6,infinity) tiling of hyperbolic plane.

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

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

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

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

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

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

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