A225499 -id:A225499 - cp's OEIS frontend

A143619 Expansion of 1/(1 - x^2 - x^7 - x^12 + x^14) (a Salem polynomial).

1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 3, 5, 5, 6, 8, 9, 12, 13, 17, 19, 24, 28, 34, 41, 49, 59, 71, 86, 103, 124, 149, 179, 215, 259, 311, 375, 450, 542, 651, 784, 942, 1133, 1363, 1638, 1971, 2369, 2851, 3427, 4123, 4957, 5962, 7170, 8622, 10370, 12470, 14998, 18035, 21691, 26085, 31371

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 26 2008

Low growth rate of 1.20262... .The absolute values of the roots of the polynomial are 0.8315201041..., 1.2026167436..., and 1.0 (with multiplicity 12). The polynomial is self-reciprocal. - Joerg Arndt, Nov 03 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,0,1,0,0,0,0,1,0,-1).

Cf. A029826, A117791, A143419, A143438, A143472, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 -x^2-x^7-x^12+x^14))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x^2 - x^7 - x^12 + x^14), {x, 0, 50}], x]
LinearRecurrence[{0,1,0,0,0,0,1,0,0,0,0,1,0,-1},{1,0,1,0,1,0,1,1,1,2,1,3,2,4},70] (* Harvey P. Dale, Aug 08 2022 *)

x='x+O('x^50); Vec(1/(1-x^2-x^7-x^12+x^14)) \\ G. C. Greubel, Nov 03 2018

G.f.: 1/(1 - x^2 - x^7 - x^12 + x^14). - Colin Barker, Nov 03 2012

a(n) = a(n-2) + a(n-7) + a(n-12) - a(n-14). - Franck Maminirina Ramaharo, Nov 02 2018

New name from Colin Barker and Joerg Arndt, Nov 03 2012

A143644 Expansion of 1/(1 - x^3 - x^4 + x^7 - x^10 - x^11 + x^14) (a Salem polynomial).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 9, 10, 12, 15, 18, 21, 26, 31, 37, 44, 54, 64, 76, 92, 111, 132, 159, 191, 229, 275, 330, 396, 475, 570, 684, 821, 985, 1182, 1418, 1703, 2043, 2451, 2942, 3531, 4236, 5084, 6101, 7321, 8785, 10543, 12652, 15182, 18219, 21864, 26237, 31485

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 26 2008

Limiting ratio is 1.2000265239873915..., the largest real root of 1 - x^3 - x^4 + x^7 - x^10 - x^11 + x^14: 1.200026523987391518902962100414 is a candidate for the smallest degree-14 Salem number. The absolute values of the roots of the polynomial are 0.8333149143..., 1.200026523..., and 1.0 (with multiplicity 12). The polynomial is self-reciprocal. - Joerg Arndt, Nov 03 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Mossinghoff, Small Salem Numbers
Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,0,-1,0,0,1,1,0,0,-1).

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/(1-x^3-x^4+x^7-x^10-x^11+x^14) )); // G. C. Greubel, Dec 06 2019

seq(coeff(series(1/(1-x^3-x^4+x^7-x^10-x^11+x^14), x, n+1), x, n), n = 0..50); # G. C. Greubel, Dec 06 2019

CoefficientList[Series[1/(1-x^3-x^4+x^7-x^10-x^11+x^14), {x, 0, 50}], x]

my(x='x+O('x^50)); Vec(1/(1-x^3-x^4+x^7-x^10-x^11+x^14)) \\ G. C. Greubel, Dec 06 2019

def A143644_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x^3-x^4+x^7-x^10-x^11+x^14) ).list()
A143644_list(50) # G. C. Greubel, Dec 06 2019

G.f.: 1/(1 - x^3 - x^4 + x^7 - x^10 - x^11 + x^14). - Colin Barker, Nov 03 2012

a(n) = a(n-3) + a(n-4) - a(n-7) + a(n-10) + a(n-11) - a(n-14). - Franck Maminirina Ramaharo, Oct 30 2018

New name from Colin Barker and Joerg Arndt, Nov 03 2012

A147663 Expansion of 1/(1 - x - x^2 + x^3 - x^7 + x^9 - x^11).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 50, 66, 88, 116, 154, 203, 269, 356, 472, 625, 828, 1097, 1453, 1925, 2550, 3379, 4476, 5930, 7855, 10406, 13784, 18260, 24189, 32044, 42449, 56233, 74493, 98682, 130726, 173175, 229409, 303902, 402585

Offset: 0

Roger L. Bagula, Nov 09 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Curtis T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, 2005.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,0,1,0,-1,0,1).

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 -x-x^2+x^3-x^7+x^9-x^11))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x - x^2 + x^3 - x^7 + x^9 - x^11), {x, 0, 50}], x] (* Franck Maminirina Ramaharo, Oct 31 2018 *)
LinearRecurrence[{1,1,-1,0,0,0,1,0,-1,0,1},{1,1,2,2,3,3,4,5,7,9,12},50] (* Harvey P. Dale, May 31 2020 *)

Vec(-1/((x^3+x^2-1)*(x^8-x^7+x^5-x^4+x^3-x+1))  + O(x^50)) \\ Colin Barker, Sep 18 2013

G.f.: -1/((x^3 + x^2 - 1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)). - Colin Barker, Sep 18 2013

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-7) - a(n-9) + a(n-10), n >= 10. - Franck Maminirina Ramaharo, Oct 31 2018

Heavily edited (because the Name, Comments, Formula and Mathematica code did not correspond to the terms of the sequence) by Colin Barker, Sep 18 2013

A173908 Expansion of 1/(1 + x - x^3 - x^4 - x^8 - x^12 - x^13 - x^17 - x^21 - x^22 - x^26 - x^30 - x^31 + x^33 + x^34).

Original entry on oeis.org

1, -1, 1, 0, 0, 0, 1, -1, 2, -2, 3, -2, 3, -2, 4, -3, 6, -5, 9, -7, 12, -9, 16, -12, 22, -17, 31, -24, 43, -33, 59, -45, 81, -63, 113, -88, 156, -121, 215, -168, 298, -233, 412, -323, 570, -448, 788, -621, 1090, -861, 1507, -1193, 2084, -1654, 2882, -2293

Offset: 0

Roger L. Bagula, Nov 26 2010

This polynomial is what I call a bi-Salem polynomial because it has two roots bigger than 1 (one positive and one negative).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,1,0,0,0,1,0,0,0,1,1,0,0,0,1,0,0,0,1,1,0,0,0,1,0,0,0,1,1,0,-1,-1).

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1+x-x^3 -x^4-x^8-x^12-x^13-x^17-x^21-x^22-x^26-x^30-x^31+x^33+x^34))); // G. C. Greubel, Nov 03 2018

seq(coeff(series(1/(1+x-x^3-x^4-x^8-x^12-x^13-x^17-x^21-x^22-x^26-x^30-x^31+ x^33+x^34), x, n+1), x, n), n = 0..60); # G. C. Greubel, Dec 15 2019

CoefficientList[Series[1/(1+x-x^3-x^4-x^8-x^12-x^13-x^17-x^21-x^22-x^26-x^30 - x^31+x^33+x^34), {x, 0, 60}], x]

x='x+O('x^60); Vec(1/(1+x-x^3-x^4-x^8-x^12-x^13-x^17-x^21-x^22-x^26 - x^30-x^31+x^33+x^34)) \\ G. C. Greubel, Nov 03 2018

def A173908_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1+x-x^3-x^4-x^8-x^12-x^13-x^17-x^21-x^22-x^26-x^30 - x^31+x^33+x^34) ).list()
A173908_list(30) # G. C. Greubel, Dec 15 2019

a(n) = a(n-1) + (n-3) + a(n-4) + a(n-8) + a(n-12) + a(n-13) + a(n-17) + a(n-21) + a(n-22) + a(n-26) + a(n-30) + a(n-31) - a(n-33) - a(n-34). - Franck Maminirina Ramaharo, Nov 02 2018

A173924 Expansion of 1/(1 - x^5 - x^6 - x^7 - x^8 + x^13).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 2, 3, 3, 3, 3, 4, 6, 8, 10, 11, 12, 16, 20, 26, 32, 38, 46, 56, 70, 88, 108, 132, 161, 198, 244, 302, 372, 457, 561, 689, 849, 1046, 1287, 1584, 1947, 2395, 2947, 3627, 4464, 5492, 6756, 8312, 10227, 12584, 15484, 19052, 23440

Offset: 0

Roger L. Bagula, Nov 26 2010

Limiting ratio is: 1.2303914344072246.
Related to the 7th Salem on the Mossinghoff's list by factorization:
(1 + x)*(1 - x + x^2)*(1 - x^3 - x^5 - x^7 + x^10)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Mossinghoff, Small Salem Numbers
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,1,1,1,0,0,0,0,-1).

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173925, A174522, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(1/(1 -x^5-x^6-x^7-x^8+x^13))); // G. C. Greubel, Nov 03 2018

seq(coeff(series(1/(1-x^5-x^6-x^7-x^8+x^13), x, n+1), x, n), n = 0..50); # G. C. Greubel, Dec 15 2019

CoefficientList[Series[1/(1-x^5-x^6-x^7-x^8+x^13), {x, 0, 50}], x]

my(x='x+O('x^50)); Vec(1/(1-x^5-x^6-x^7-x^8+x^13)) \\ G. C. Greubel, Nov 03 2018

def A173924_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x^5-x^6-x^7-x^8+x^13) ).list()
A173924_list(50) # G. C. Greubel, Dec 15 2019

a(n) = a(n-5) + a(n-6) + a(n-7) + a(n-8) - a(n-13). - Franck Maminirina Ramaharo, Oct 30 2018

More terms from Franck Maminirina Ramaharo, Nov 03 2018

A173925 Expansion of 1/(1 - x - x^8 - x^15 + x^16).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 19, 24, 30, 37, 45, 56, 69, 85, 105, 130, 161, 199, 246, 304, 376, 465, 575, 711, 879, 1086, 1343, 1660, 2052, 2537, 3137, 3879, 4796, 5929, 7330, 9062, 11203, 13850, 17123, 21170, 26173, 32359, 40006

Offset: 0

Roger L. Bagula, Nov 26 2010

Limiting ratio is 1.2303914344072246.

The polynomial is the 10th Salem on Mossinghoff's list.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Mossinghoff, Small Salem Numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1).

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A174522, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!(1/(1-x-x^8-x^15+x^16))); // G. C. Greubel, Nov 03 2018

seq(coeff(series(1/(1-x-x^8-x^15+x^16), x, n+1), x, n), n = 0..60); # G. C. Greubel, Dec 15 2019

CoefficientList[Series[1/(1-x-x^8-x^15+x^16), {x, 0, 60}] ,x] (* Harvey P. Dale, Apr 02 2012 *)

my(x='x+O('x^60)); Vec(1/(1-x-x^8-x^15+x^16)) \\ G. C. Greubel, Nov 03 2018

def A173925_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x-x^8-x^15+x^16) ).list()
A173925_list(60) # G. C. Greubel, Dec 15 2019

a(n) = a(n-1) + a(n-8) + a(n-15) - a(n-16). - Harvey P. Dale, Apr 02 2012

A174522 Expansion of 1/(1 - x - x^4 + x^6).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 3, 4, 6, 7, 7, 8, 11, 14, 15, 16, 20, 26, 30, 32, 37, 47, 57, 63, 70, 85, 105, 121, 134, 156, 191, 227, 256, 291, 348, 419, 484, 548, 640, 768, 904, 1033, 1189, 1409, 1673, 1938, 2223, 2599, 3083, 3612, 4162, 4823, 5683, 6696, 7775, 8986

Offset: 0

Roger L. Bagula, Nov 28 2010

Low limiting ratio in 100th iteration near 1.16663.

The polynomial is interesting for the puzzling low ratio and the Salem like root structure with two complex roots outside the unit circle.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,0,-1).

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A175740, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 - x - x^4 + x^6))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x - x^4 + x^6), {x, 0, 60}], x]

x='x+O('x^50); Vec(1/(1 - x - x^4 + x^6)) \\ G. C. Greubel, Nov 03 2018

a(n) = a(n-1) + a(n-4) + a(n-6). - Franck Maminirina Ramaharo, Oct 31 2018

A175740 Expansion of 1/(1 - x - x^10 - x^19 + x^20).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 21, 26, 32, 39, 47, 56, 66, 79, 94, 112, 134, 161, 194, 234, 282, 339, 407, 488, 585, 701, 840, 1007, 1208, 1450, 1741, 2090, 2510, 3013, 3616, 4339, 5206, 6246, 7494, 8992, 10790, 12948

Offset: 0

Roger L. Bagula, Dec 04 2010

Limiting ratio is 1.2000265239873915.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Mossinghoff, Small Salem Numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1).

Cf. A175739.

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175772, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!(1/(1 - x - x^10 - x^19 + x^20))); // G. C. Greubel, Nov 03 2018

seq(coeff(series(1/(1 -x -x^10 -x^19 +x^20), x, n+1), x, n), n = 0..60); # G. C. Greubel, Dec 05 2019

CoefficientList[Series[1/(1 -x -x^10 -x^19 +x^20), {x, 0, 60}], x]

my(x='x+O('x^60)); Vec(1/(1 -x -x^10 -x^19 +x^20)) \\ G. C. Greubel, Nov 03 2018

def A175740_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1 -x -x^10 -x^19 +x^20) ).list()
A175740_list(60) # G. C. Greubel, Dec 05 2019

G.f.: 1/((1 - x + x^2)*(1 - x^2 + x^4)*(1 - x^3 - x^4 + x^7 - x^10 - x^11 + x^14)).

a(n) = a(n-1) + a(n-10) + a(n-19) + a(n-20). - Franck Maminirina Ramaharo, Oct 31 2018

A175772 Expansion of 1/(1 - x - x^9 - x^17 + x^18).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 16, 20, 25, 31, 38, 46, 55, 67, 81, 98, 119, 145, 177, 216, 263, 320, 389, 473, 575, 699, 850, 1034, 1258, 1530, 1862, 2265, 2755, 3351, 4076, 4958, 6031, 7336, 8923, 10854, 13203, 16060, 19535, 23762

Offset: 0

Roger L. Bagula, Dec 04 2010

The ratio a(n+1)/a(n) is 1.216391661138265... as n->infinity.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Mossinghoff, Small Salem Numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1).

Cf. A175739.

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175773, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^9-x^17+x^18))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x - x^9 - x^17 + x^18), {x, 0, 50}], x] (* or *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11} ,60] (* Harvey P. Dale, Jul 13 2014 *)

x='x+O('x^50); Vec(1/(1-x-x^9-x^17+x^18)) \\ G. C. Greubel, Nov 03 2018

G.f.: 1/((1 - x^2 + x^4)*(1 - x^4 - x^5 - x^6 + x^10)*(1 - x + x^2 - x^3 + x^4)).

a(n) = a(n-1) + a(n-9) + a(n-17) - a(n-18). - Harvey P. Dale, Jul 13 2014

A175773 Expansion of 1/(1 - x - x^6 - x^11 + x^12).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 10, 13, 17, 22, 28, 37, 48, 62, 80, 103, 133, 172, 223, 289, 374, 483, 625, 808, 1045, 1352, 1749, 2262, 2926, 3785, 4896, 6333, 8191, 10595, 13704, 17726, 22929, 29659, 38363, 49622, 64185, 83022, 107388, 138905, 179672

Offset: 0

Roger L. Bagula, Dec 04 2010

The ratio a(n+1)/a(n) is 1.2934859531254534... for n->infinity.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Mossinghoff, Small Salem numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,0,0,0,0,1,-1).

Cf. A175739.

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175782, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^6-x^11+x^12))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x - x^6 - x^11 + x^12), {x, 0, 50}], x]

x='x+O('x^50); Vec(1/(1-x-x^6-x^11+x^12)) \\ G. C. Greubel, Nov 03 2018

G.f.: 1/((1 - x + x^2)*(1 - x^2 - x^3 + x^5 - x^7 - x^8 + x^10)).

a(n) = a(n-1) + a(n-6) + a(n-11) - a(n-12), n >= 12. - Franck Maminirina Ramaharo, Oct 31 2018

cp's OEIS Frontend

A143619 Expansion of 1/(1 - x^2 - x^7 - x^12 + x^14) (a Salem polynomial).

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A143644 Expansion of 1/(1 - x^3 - x^4 + x^7 - x^10 - x^11 + x^14) (a Salem polynomial).

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A147663 Expansion of 1/(1 - x - x^2 + x^3 - x^7 + x^9 - x^11).

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A173908 Expansion of 1/(1 + x - x^3 - x^4 - x^8 - x^12 - x^13 - x^17 - x^21 - x^22 - x^26 - x^30 - x^31 + x^33 + x^34).

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A173924 Expansion of 1/(1 - x^5 - x^6 - x^7 - x^8 + x^13).

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A173925 Expansion of 1/(1 - x - x^8 - x^15 + x^16).

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Magma

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