A001630 -id:A001630 - cp's OEIS frontend

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

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

A074584 Esanacci (hexanacci or "6-anacci") numbers.

Original entry on oeis.org

6, 1, 3, 7, 15, 31, 63, 120, 239, 475, 943, 1871, 3711, 7359, 14598, 28957, 57439, 113935, 225999, 448287, 889215, 1763832, 3498707, 6939975, 13766015, 27306031, 54163775, 107438335, 213112838, 422726969, 838513963, 1663261911, 3299217791, 6544271807

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 26 2002

These esanacci numbers follow the same pattern as Lucas, generalized tribonacci (A001644), generalized tetranacci (A073817), and generalized pentanacci (A074048) numbers.
The closed form is a(n) = r1^n + r^2^n + r3^n + r4^n + r5^n + r6^n, with r1, r2, r3, r4, r5, r6 roots of the characteristic polynomial.
a(n) is also the trace of A^n, where A is the matrix ((1, 1, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0), (1, 0, 0, 1, 0, 0), (1, 0, 0, 0, 1, 0), (1, 0, 0, 0, 0, 1), (1, 0, 0, 0, 0, 0)).

T. D. Noe, Table of n, a(n) for n=0..200
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
Spiros D. Dafnis, Andreas N. Philippou, and Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1).

Cf. A000078, A001630, A001644, A000032, A073817, A074048.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6) )); // G. C. Greubel, Apr 22 2019

CoefficientList[Series[(6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6), {x, 0, 40}], x]
LinearRecurrence[{1,1,1,1,1,1},{6,1,3,7,15,31},40] (* Harvey P. Dale, Nov 08 2011 *)

polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6), 40) \\ G. C. Greubel, Apr 22 2019

def aupton(nn):
  alst = [6, 1, 3, 7, 15, 31]
  for n in range(6, nn+1): alst.append(sum(alst[n-6:n]))
  return alst[:nn+1]
print(aupton(33)) # Michael S. Branicky, Jun 01 2021

((6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6), a(0)=6, a(1)=1, a(2)=3, a(3)=7, a(4)=15, a(5)=31.
G.f.: (6-5*x-4*x^2-3*x^3-2*x^4-x^5)/(1-x-x^2-x^3-x^4-x^5-x^6).
a(n) = 2*a(n-1) - a(n-7) for n >= 7. - Vincenzo Librandi, Dec 20 2010

A249413 Primes in the hexanacci numbers sequence A000383.

Original entry on oeis.org

11, 41, 72426721, 143664401, 565262081, 4160105226881, 253399862985121, 997027328131841, 212479323351825962211841, 188939838859312612896128881921, 22828424707602602744356458636161, 661045104283639247572028952777478721

Offset: 1

Robert Price, Dec 03 2014

nonn

a(13) is too large to display here. It has 62 digits and is the 210th term in A000383.

Cf. A001590, A001631, A100683, A231574, A231575, A232543, A214899, A020992, A233554, A214727, A234696, A141523, A235862, A214825, A235873, A001630, A241660, A247027, A000288, A247561, A000322, A248920, A000383, A247192.

a={1,1,1,1,1,1}; For[n=6, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[sum]]; a=RotateLeft[a]; a[[5]]=sum]

A251656 4-step Fibonacci sequence starting with 1,0,1,0.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 6, 11, 22, 42, 81, 156, 301, 580, 1118, 2155, 4154, 8007, 15434, 29750, 57345, 110536, 213065, 410696, 791642, 1525939, 2941342, 5669619, 10928542, 21065442, 40604945, 78268548, 150867477, 290806412, 560547382, 1080489819

Offset: 0

Arie Bos, Dec 06 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251703, A251704, A251705.

Cf. A000336.

NB. see A251655 for the program and apply it to 1,0,1,0.

LinearRecurrence[Table[1, {4}], {1, 0, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n)+a(n+1)+a(n+2)+a(n+3).
G.f.: (-1+x+2*x^3)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+3)-A000078(n+2)-2*A000078(n). - R. J. Mathar, Mar 28 2025

A251703 4-step Fibonacci sequence starting with 1,1,0,0.

Original entry on oeis.org

1, 1, 0, 0, 2, 3, 5, 10, 20, 38, 73, 141, 272, 524, 1010, 1947, 3753, 7234, 13944, 26878, 51809, 99865, 192496, 371048, 715218, 1378627, 2657389, 5122282, 9873516, 19031814, 36685001, 70712613, 136302944, 262732372, 506432930, 976180859

Offset: 0

Arie Bos, Dec 07 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251656, A251704, A251705.

NB. see A251655 for the program and apply it to 1,1,0,0.

LinearRecurrence[Table[1, {4}], {1, 1, 0, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: (-1+2*x^2+2*x^3)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+3)-2*A000078(n+1)-2*A000078(n). - R. J. Mathar, Mar 28 2025

A251654 4-step Fibonacci sequence starting with 0, 1, 1, 0.

Original entry on oeis.org

0, 1, 1, 0, 2, 4, 7, 13, 26, 50, 96, 185, 357, 688, 1326, 2556, 4927, 9497, 18306, 35286, 68016, 131105, 252713, 487120, 938954, 1809892, 3488679, 6724645, 12962170, 24985386, 48160880, 92833081, 178941517, 344920864, 664856342, 1281551804

Offset: 0

Arie Bos, Dec 06 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251655, A251656, A251672, A251703, A251704, A251705.

NB. see A251655 for the program and apply it to 0,1,1,0.

LinearRecurrence[Table[1, {4}], {0, 1, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: x*(-1+2*x^2)/(-1+x+x^2+x^3+x^4). - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+2)-2*A000078(n). - R. J. Mathar, Mar 28 2025

A251655 4-step Fibonacci sequence starting with 0, 1, 1, 1.

Original entry on oeis.org

0, 1, 1, 1, 3, 6, 11, 21, 41, 79, 152, 293, 565, 1089, 2099, 4046, 7799, 15033, 28977, 55855, 107664, 207529, 400025, 771073, 1486291, 2864918, 5522307, 10644589, 20518105, 39549919, 76234920, 146947533, 283250477, 545982849, 1052415779, 2028596638

Offset: 0

Arie Bos, Dec 06 2014

Tamás Lengyel and Diego Marques, The 2-adic Order of Some Generalized Fibonacci Numbers, INTEGERS, 17, 2017, A5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251656, A251672, A251703, A251704, A251705.

(see www.jsoftware.com) First construct the generating matrix
   [M=: (#.@}: + {:)\"1&.|: <:/~i.4
1 1 1 1
1 2 2 2
2 3 4 4
4 6 7 8
Given that matrix, one can produce the first 4*250 numbers with
, M(+/ . *)^:(i.250) 0 1 1 1x

LinearRecurrence[Table[1, {4}], {0, 1, 1, 1}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: x*(x-1)*(1+x)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+2)-A000078(n). - R. J. Mathar, Mar 28 2025

A251704 4-step Fibonacci sequence starting with 1, 1, 0, 1.

Original entry on oeis.org

1, 1, 0, 1, 3, 5, 9, 18, 35, 67, 129, 249, 480, 925, 1783, 3437, 6625, 12770, 24615, 47447, 91457, 176289, 339808, 655001, 1262555, 2433653, 4691017, 9042226, 17429451, 33596347, 64759041, 124827065, 240611904, 463794357, 893992367, 1723225693

Offset: 0

Arie Bos, Dec 07 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251656, A251703, A251705.

NB. see A251655 for the program and apply it to 1,1,0,1.

LinearRecurrence[Table[1, {4}], {1, 1, 0, 1}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: (1+x)*(x^2+x-1)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A001630(n-2)+A001630(n-1), n>2. - R. J. Mathar, Mar 28 2025

cp's OEIS Frontend

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

A074584 Esanacci (hexanacci or "6-anacci") numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Sage

Formula

A249413 Primes in the hexanacci numbers sequence A000383.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A251656 4-step Fibonacci sequence starting with 1,0,1,0.

Views

Author

Keywords

Links

Crossrefs

Programs

J

Mathematica

Formula

A251703 4-step Fibonacci sequence starting with 1,1,0,0.

Views

Author

Keywords

Links

Crossrefs

Programs

J

Mathematica

Formula

A251654 4-step Fibonacci sequence starting with 0, 1, 1, 0.

Views

Author

Keywords

Links

Crossrefs