A117791 -id:A117791 - cp's OEIS frontend

A173696 a(n) = ceiling(A117791(n)/2).

1, 1, 1, 2, 2, 3, 5, 7, 10, 15, 23, 34, 51, 77, 115, 173, 259, 389, 584, 877, 1317, 1978, 2970, 4459, 6696, 10054, 15097, 22670, 34041, 51115, 76754, 115254, 173064, 259872, 390223, 585956, 879869, 1321206, 1983916, 2979038, 4473308, 6717096, 10086357, 15145621, 22742585, 34150148, 51279684, 77001307

Offset: 0

Roger L. Bagula, Nov 25 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A117791.

Ceiling[CoefficientList[Series[1/(1 - x - x^2 + x^4 - x^6), {x, 0, 50}], x]/2] (* G. C. Greubel, Nov 16 2016 *)

a(n) = A117791(n) - floor(A117791(n)/2) = ceiling(A117791(n)/2).

Erroneous leading zeros deleted by Joerg Arndt, Nov 17 2016

Sequence values corrected by G. C. Greubel, Nov 19 2016

A204631 Expansion of 1/(1 - x - x^2 + x^5 - x^7).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 17, 26, 40, 62, 96, 148, 229, 354, 547, 845, 1306, 2018, 3118, 4818, 7445, 11504, 17776, 27468, 42444, 65585, 101343, 156597, 241976, 373905, 577764, 892770, 1379522, 2131659, 3293873, 5089744, 7864752, 12152738, 18778601, 29016988

Offset: 0

Roger L. Bagula, Jan 17 2012

Limiting ratio is 1.5452156..., the real root of x^7 - x^6 - x^5 + x^2 - 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 8.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,1).

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^2+x^5-x^7))); // G. C. Greubel, Nov 03 2018

seq(coeff(series(1/(1-x-x^2+x^5-x^7), x, n+1), x, n), n = 0..50); # G. C. Greubel, Mar 16 2020

CoefficientList[Series[1/(1 - x - x^2 + x^5 - x^7), {x, 0, 50}], x]
LinearRecurrence[{1,1,0,0,-1,0,1},{1,1,2,3,5,7,11},50] (* Harvey P. Dale, Aug 28 2013 *)

my(x='x+O('x^50)); Vec(1/(1-x-x^2+x^5-x^7)) \\ G. C. Greubel, Nov 16 2016

a(n) = a(n-1) + a(n-2) - a(n-5) + a(n-7). - Franck Maminirina Ramaharo, Nov 02 2018

A225391 Expansion of 1/(1 - x - x^2 - x^6 + x^8).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 23, 38, 63, 104, 172, 285, 472, 781, 1293, 2140, 3542, 5863, 9705, 16064, 26590, 44013, 72852, 120588, 199603, 330392, 546880, 905221, 1498363, 2480159, 4105273, 6795236, 11247786, 18617851, 30817120, 51009909, 84433939, 139758925

Offset: 0

Roger L. Bagula, May 06 2013

Limiting ratio is 1.65525..., the largest real root of 1 - x^2 - x^6 - x^7 + x^8.

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

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^2-x^6+x^8))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x - x^2 - x^6 + x^8), {x, 0, 50}], x]
LinearRecurrence[{1,1,0,0,0,1,0,-1}, {1,1,2,3,5,8,14,23}, 100] (* G. C. Greubel, Nov 16 2016 *)

Vec(1/(1-x-x^2-x^6+x^8) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

a(n) = a(n-1) + a(n-2) + a(n-6) - a(n-8). - Franck Maminirina Ramaharo, Nov 02 2018

A225482 Expansion of 1/(1 - x^3 - x^4 - x^5 + x^8).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 4, 6, 8, 10, 12, 16, 21, 26, 34, 43, 55, 71, 91, 116, 148, 191, 244, 312, 400, 512, 656, 840, 1076, 1377, 1764, 2260, 2893, 3705, 4745, 6077, 7782, 9966, 12763, 16344, 20932, 26806, 34328, 43962, 56300, 72100, 92333, 118246

Offset: 0

Roger L. Bagula, May 08 2013

Limiting ratio is 1.28064..., the largest real root of 1 - x^3 - x^4 - x^5 + x^8: 1.280638156267757596701902532710 is a candidate for the smallest degree-8 Salem number.

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 (0,0,1,1,1,0,0,-1).

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^3-x^4-x^5+x^8))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x^3 - x^4 - x^5 + x^8), {x, 0, 50}], x]
LinearRecurrence[{0,0,1,1,1,0,0,-1}, {1,0,0,1,1,1,1,2}, 100] (* G. C. Greubel, Nov 16 2016 *)

Vec(1/(1-x^3-x^4-x^5+x^8)+O(x^99)) \\ Charles R Greathouse IV, May 08 2013

a(n) = a(n-3) + a(n-4) + a(n-5) - a(n-8). - Franck Maminirina Ramaharo, Nov 02 2018

More terms from Franck Maminirina Ramaharo, Nov 02 2018

A143419 G.f.: 1/p(x), where p(x) = degree 22 Salem polynomial p(x) = x^22 + x^21 - x^19 - 2x^18 - 3x^17 - 3x^16 - 2x^15 + 2x^13 + 4x^12 + 5x^11 + 4x^10 + 2x^9 - 2x^7 - 3x^6 - 3x^5 - 2*x^4 - x^3 + x + 1.

Original entry on oeis.org

1, -1, 1, 0, 1, 1, 1, 2, 2, 4, 4, 7, 9, 12, 17, 23, 32, 44, 60, 83, 113, 156, 214, 294, 403, 554, 760, 1044, 1433, 1967, 2701, 3708, 5091, 6988, 9596, 13172, 18085, 24828, 34086, 46797, 64246, 88203, 121092, 166246, 228237, 313343, 430185, 590594, 810819

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 23 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, 2001
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,2,3,3,2,0,-2,-4,-5,-4,-2,0,2,3,3,2,1,0,-1,-1).

Cf. A029826, A117791, A143438, A143472, A143619, 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/(x^22 +x^21-x^19-2*x^18-3*x^17-3*x^16-2*x^15+2*x^13+4*x^12+5*x^11 + 4*x^10+2*x^9-2*x^7-3*x^6-3*x^5-2*x^4-x^3+x+1))); // G. C. Greubel, Nov 03 2018

f[x_] = x^22 + x^21 - x^19 - 2*x^18 - 3*x^17 - 3*x^16 - 2*x^15 + 2*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 2*x^9 - 2*x^7 - 3*x^6 - 3*x^5 - 2*x^4 - x^3 + x + 1;
CoefficientList[Series[1/f[x], {x, 0, 50}], x]
LinearRecurrence[{-1,0,1,2,3,3,2,0,-2,-4,-5,-4,-2,0,2,3,3,2,1,0,-1,-1},{1,-1,1,0,1,1,1,2,2,4,4,7,9,12,17,23,32,44,60,83,113,156},50] (* Harvey P. Dale, Aug 18 2024 *)

p(x)=x^22 + x^21 - x^19 - 2*x^18 - 3*x^17 - 3*x^16 - 2*x^15 + 2*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 2*x^9 - 2*x^7 - 3*x^6 - 3*x^5 - 2*x^4 - x^3 + x + 1; Vec(1/p(x)+O(x^60)) \\ Charles R Greathouse IV, Feb 13 2011

a(n) = -a(n-1) + a(n-3) + 2*a(n-4) + 3*a(n-5) + 3*a(n-6) + 2*a(n-7) - 2*a(n-9) - 4*a(n-10) - 5*a(n-11) - 4*a(n-12) - 2*a(n-13) + 2*a(n-15) + 3*a(n-16) + 3*a(n-17) + 2*a(n-18) + a(n-19) - a(n-21) - a(n-22). - Franck Maminirina Ramaharo, Oct 30 2018

Edited by N. J. A. Sloane, Dec 12 2008
More terms from Sean A. Irvine, Feb 13 2011
Offset corrected, and more terms from Franck Maminirina Ramaharo, Nov 02 2018

A173911 Expansion of 1/(1 - x + x^2 - x^3 - x^6 + x^7 - x^8 + x^9 - x^10 + x^11 - x^12 -x^15 + x^16 - x^17 + x^18).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 12, 14, 16, 19, 23, 28, 33, 39, 46, 55, 66, 78, 92, 110, 131, 155, 184, 219, 260, 309, 368, 437, 519, 617, 733, 871, 1036, 1231, 1462, 1737, 2065, 2454, 2916, 3465, 4118, 4894, 5816, 6911, 8213

Offset: 0

Roger L. Bagula, Nov 26 2010

Limiting ratio is 1.188368147508223588... = A219300.

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,-1,1,0,0,1,-1,1,-1,1,-1,1,0,0,1,-1,1,-1).

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

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(1/(1-x+x^2-x^3-x^6+x^7-x^8+x^9-x^10+x^11-x^12-x^15+x^16-x^17+x^18))); // G. C. Greubel, Nov 03 2018

seq(coeff(series(1/(1-x+x^2-x^3-x^6+x^7-x^8+x^9-x^10+x^11-x^12-x^15+x^16 -x^17+x^18), x, n+1), x, n), n = 0..50); # G. C. Greubel, Dec 15 2019

CoefficientList[Series[1/(1-x+x^2-x^3-x^6+x^7-x^8+x^9-x^10+x^11-x^12-x^15+x^16 -x^17+x^18), {x,0,50}], x]
LinearRecurrence[{1,-1,1,0,0,1,-1,1,-1,1,-1,1,0,0,1,-1,1,-1},{1,1,0,0,1,1,1,1,1,1,2,2,2,3,4,4,5,6},60] (* Harvey P. Dale, Apr 02 2024 *)

my(x='x+O('x^50)); Vec(1/(1-x+x^2-x^3-x^6+x^7-x^8+x^9-x^10+x^11-x^12-x^15+ x^16-x^17+x^18)) \\ G. C. Greubel, Nov 03 2018

def A173911_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-x+x^2-x^3-x^6+x^7-x^8+x^9-x^10+x^11-x^12-x^15+x^16 -x^17+x^18) ).list()
A173911_list(50) # G. C. Greubel, Dec 15 2019

a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-6) - a(n-7) + a(n-8) - a(n-9) + a(n-10) - a(n-11) + a(n-12) + a(n-15) - a(n-16) + a(n-17) - a(n-16). - Franck Maminirina Ramaharo, Nov 02 2018

A225393 Expansion of 1/(1 - x - x^2 + x^6 - x^8).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 19, 30, 47, 74, 116, 183, 288, 453, 713, 1122, 1766, 2779, 4373, 6882, 10830, 17043, 26820, 42206, 66419, 104522, 164484, 258845, 407339, 641021, 1008761, 1587466, 2498162, 3931305, 6186612, 9735741, 15320931, 24110227, 37941757, 59708145

Offset: 0

Roger L. Bagula, May 06 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 8.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,0,-1,0,1).

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

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^2+x^6-x^8))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x - x^2 + x^6 - x^8), {x, 0, 50}], x]
LinearRecurrence[{1,1,0,0,0,-1,0,1},{1,1,2,3,5,8,12,19},50] (* G. C. Greubel, Nov 16 2016 *)

Vec(1/(1-x-x^2+x^6-x^8) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

G.f.: 1/(1 - x - x^2 + x^6 - x^8).

a(n) = a(n-1) + a(n-2) - a(n-6) + a(n-8). - Ilya Gutkovskiy, Nov 16 2016

A225394 Expansion of 1/(1 - x - x^2 + x^7 - x^9).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 20, 32, 51, 81, 129, 205, 326, 519, 826, 1314, 2091, 3327, 5294, 8424, 13404, 21328, 33937, 54000, 85924, 136721, 217548, 346159, 550803, 876429, 1394560, 2219002, 3530841, 5618219, 8939622, 14224586, 22633938, 36014767, 57306132, 91184618

Offset: 0

Roger L. Bagula, May 06 2013

Limiting ratio is 1.59118..., the largest real root of -1 + x^2 - x^7 - x^8 + x^9 = 0.

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

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

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

CoefficientList[Series[1/(1 - x - x^2 + x^7 - x^9), {x, 0, 50}], x]
LinearRecurrence[{1,1,0,0,0,0,-1,0,1}, {1,1,2,3,5,8,13,20,32}, 50] (* G. C. Greubel, Nov 16 2016 *)

Vec(1/(1-x-x^2+x^7-x^9) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

a(n) = a(n-1) + a(n-2) - a(n-7) + a(n-9). - Ilya Gutkovskiy, Nov 16 2016

A225499 Expansion of 1/(1 - x + x^2 - x^3 - x^6 - x^9 + x^10 - x^11 + x^12).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 7, 9, 11, 13, 16, 20, 25, 31, 39, 48, 59, 74, 92, 113, 140, 175, 217, 269, 334, 414, 513, 637, 791, 981, 1217, 1510, 1874, 2325, 2884, 3578, 4440, 5509, 6835, 8481, 10522, 13054, 16197, 20097, 24934, 30936, 38384

Offset: 0

Roger L. Bagula, May 09 2013

Limiting ratio is 1.24073..., the largest real root of 1 - x + x^2 - x^3 - x^6 - x^9 + x^10 - x^11 + x^12: 1.240726423652541392056148161575

is a candidate for the smallest degree-12 Salem number.

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,-1,1,0,0,1,0,0,1,-1,1,-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.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 -x +x^2 -x^3 -x^6 -x^9 +x^10 -x^11 +x^12))); // G. C. Greubel, Nov 03 2018

CoefficientList[Series[1/(1 - x + x^2 - x^3 - x^6 - x^9 + x^10 - x^11 + x^12), {x, 0, 50}], x]
LinearRecurrence[{1,-1,1,0,0,1,0,0,1,-1,1,-1},{1,1,0,0,1,1,1,2,2,2,3,4},100] (* G. C. Greubel, Nov 16 2016 *)

Vec(1/(1 -x +x^2 -x^3 -x^6 -x^9 +x^10 -x^11 +x^12) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-6) + a(n-9) - a(n-10) + a(n-11) - a(n-12). - Franck Maminirina Ramaharo, Nov 02 2018

A143438 a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6), with a(0) = a(2) = a(3) = 1, a(1) = 0 and a(4) = a(5) = 2.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 5, 6, 9, 12, 18, 24, 34, 48, 67, 94, 131, 185, 258, 362, 507, 711, 996, 1395, 1956, 2740, 3840, 5380, 7540, 10565, 14804, 20745, 29069, 40734, 57078, 79983, 112077, 157050, 220069, 308376, 432118, 605512, 848486, 1188956, 1666047, 2334578

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 23 2008

Expansion of 1/p(x), where p(x) = 1 - x^2 - x^3 - x^4 + x^6 is a Salem polynomial.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Curtis T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, 2001.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1).

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

a:=[1,0,1,1,2,2];; for n in [7..50] do a[n]:=a[n-2]+a[n-3]+a[n-4]-a[n-6]; od; a; # G. C. Greubel, Dec 06 2019

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!(1/(1 -x^2-x^3-x^4+x^6))); // G. C. Greubel, Nov 03 2018

seq(coeff(series(1/(1-x^2-x^3-x^4+x^6), x, n+1), x, n), n = 0..50); # G. C. Greubel, Dec 06 2019

CoefficientList[Series[1/(1-x^2-x^3-x^4+x^6), {x, 0, 50}], x]
LinearRecurrence[{0,1,1,1,0,-1}, {1,0,1,1,2,2}, 50] (* G. C. Greubel, Dec 06 2019 *)

makelist(ratcoef(taylor(1/(1 -x^2 -x^3 -x^4 +x^6), x, 0, n), x, n), n, 0, 50); /* Franck Maminirina Ramaharo, Oct 31 2018 */

my(x='x+O('x^50)); Vec(1/(1-x^2-x^3-x^4+x^6)) \\ G. C. Greubel, Nov 03 2018

def A143438_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/(1-x^2-x^3-x^4+x^6)).list()
A143438_list(50) # G. C. Greubel, Dec 06 2019

G.f.: 1/(1 - x^2 - x^3 - x^4 + x^6). - Colin Barker, Nov 24 2012

Edited, new name (after Colin Barker), more terms, and offset corrected by Franck Maminirina Ramaharo, Oct 30 2018

cp's OEIS Frontend

A173696 a(n) = ceiling(A117791(n)/2).

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A204631 Expansion of 1/(1 - x - x^2 + x^5 - x^7).

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A225391 Expansion of 1/(1 - x - x^2 - x^6 + x^8).

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A225482 Expansion of 1/(1 - x^3 - x^4 - x^5 + x^8).

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A143419 G.f.: 1/p(x), where p(x) = degree 22 Salem polynomial p(x) = x^22 + x^21 - x^19 - 2*x^18 - 3*x^17 - 3*x^16 - 2*x^15 + 2*x^13 + 4*x^12 + 5*x^11 + 4*x^10 + 2*x^9 - 2*x^7 - 3*x^6 - 3*x^5 - 2*x^4 - x^3 + x + 1.

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A173911 Expansion of 1/(1 - x + x^2 - x^3 - x^6 + x^7 - x^8 + x^9 - x^10 + x^11 - x^12 -x^15 + x^16 - x^17 + x^18).

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A225393 Expansion of 1/(1 - x - x^2 + x^6 - x^8).

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A143419 G.f.: 1/p(x), where p(x) = degree 22 Salem polynomial p(x) = x^22 + x^21 - x^19 - 2x^18 - 3x^17 - 3x^16 - 2x^15 + 2x^13 + 4x^12 + 5x^11 + 4x^10 + 2x^9 - 2x^7 - 3x^6 - 3x^5 - 2*x^4 - x^3 + x + 1.