A029826 -id:A029826 - cp's OEIS frontend

A172497 Triangle T(n, k) = round( c(n)/(c(k)*c(n-k)) ) where c(n) = Product_{j=1..n} A029826(j+10), read by rows.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 4, 2, 2, 1, 1, 3, 6, 6, 6, 6, 3, 1, 1, 2, 6, 12, 6, 12, 6, 2, 1, 1, 4, 8, 24, 24, 24, 24, 8, 4, 1, 1, 3, 12, 24, 36, 72, 36, 24, 12, 3, 1, 1, 5, 15, 60, 60, 180, 180, 60, 60, 15, 5, 1, 1, 5, 25, 75, 150, 300, 450, 300, 150, 75, 25, 5, 1

Offset: 0

Roger L. Bagula, Feb 05 2010

			The triangle begins as:
  1;
  1, 1;
  1, 1,  1;
  1, 1,  1,  1;
  1, 2,  2,  2,  1;
  1, 1,  2,  2,  1,  1;
  1, 2,  2,  4,  2,  2,  1;
  1, 3,  6,  6,  6,  6,  3,  1;
  1, 2,  6, 12,  6, 12,  6,  2,  1;
  1, 4,  8, 24, 24, 24, 24,  8,  4, 1;
  1, 3, 12, 24, 36, 72, 36, 24, 12, 3, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A029826.

R:= PowerSeriesRing(Integers(), 100);
b:= Coefficients(R!( 1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10) ));
c:= func< n | (&*[b[j]: j in [10..n+10]]) >;
T:= func< n,k | Round(c(n)/(c(k)*c(n-k))) >;
[T(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, Apr 20 2021

b:= Drop[CoefficientList[Series[1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10), {x,0,100}], x], 10];
c[n_]:= Product[b[[j]], {j,n}];
T[n_, k_]:= Round[c[n]/(c[k]*c[n-k])];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 20 2021 *)

@CachedFunction
def A029826_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10) ).list()
b=A029826_list(130)
def c(n): return product(b[j] for j in (9..n+9))
def T(n,k): return round(c(n)/(c(k)*c(n-k)))
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 20 2021

T(n, k) = round( c(n)/(c(k)*c(n-k)) ) where c(n) = Product_{j=1..n} A029826(j+10).

Definition corrected and edited by G. C. Greubel, Apr 20 2021

A173894 a(n) = ceiling(A029826(n)/2).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 14, 16, 19, 22, 25, 31, 35, 42, 49, 58, 68, 80, 94, 110, 130, 152, 180, 210, 248, 292, 343, 404, 474, 558, 656, 772, 908, 1068, 1256, 1478, 1738, 2045, 2406, 2829, 3328, 3914, 4605, 5416, 6371, 7494, 8815, 10369, 12197, 14347

Offset: 0

Roger L. Bagula, Nov 26 2010

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

Cf. A029826.

R:= PowerSeriesRing(Integers(), 105);
A029826:= Coefficients(R!( 1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10) ));
A173894:= func< n | Ceiling( A029826[n+1]/2 ) >;
[A173894(n): n in [0..100]]; // G. C. Greubel, Apr 23 2021

A029826 = CoefficientList[Series[1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10), {x, 0, 250}], x];
Table[Ceiling[A029826[[n+1]]/2], {n, 0, 100}] (* modified by G. C. Greubel, Apr 23 2021 *)

A029826=[( 1/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10) ).series(x,n+1).list()[n] for n in (0..100)]
def A173894(n): return ceil( A029826[n]/2 )
[A173894(n) for n in (0..100)] # G. C. Greubel, Apr 23 2021

a(n) = ceiling(A029826(n)/2) = A029826(n) - floor(A029826(n)/2).

A172972 Subtraction triangle based on A029826: c(n)=Product[A029826(i),{i,0,n)];t(n,m)=c(n)-c(m)-c(n-m).

Original entry on oeis.org

-1, -1, -1, -1, -3, -1, -1, -1, -1, -1, -1, 0, 2, 0, -1, -1, -1, 2, 2, -1, -1, -1, -1, 1, 2, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 0, 1, 1, -1, -1, -1, -1, 1, 1, 0, 0, 1, 1, -1, -1, -1, -1, 1, 1, 0, 0, 0, 1, 1, -1, -1

Offset: 0

Roger L. Bagula, Feb 06 2010

Row sums are:

{-1, -2, -5, -4, 0, 0, 0, 0, 0, 0, 0,...}.

			{-1},
{-1, -1},
{-1, -3, -1},
{-1, -1, -1, -1},
{-1, 0, 2, 0, -1},
{-1, -1, 2, 2, -1, -1},
{-1, -1, 1, 2, 1, -1, -1},
{-1, -1, 1, 1, 1, 1, -1, -1},
{-1, -1, 1, 1, 0, 1, 1, -1, -1},
{-1, -1, 1, 1, 0, 0, 1, 1, -1, -1},
{-1, -1, 1, 1, 0, 0, 0, 1, 1, -1, -1}

A029826

(*A029826 Inverse of Salem polynomial : 1/(x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1).*)
p[x_] = (x^(10) + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1); q[ x_] = Expand[x^10*p[1/x]]; a = Table[SeriesCoefficient[Series[1/ q[x], {x, 0, 100}], n], {n, 0, 100}];
c[n_] := Product[a[[m]], {m, 1, n}];
t[n_, m_] := c[n] - (c[m] + c[n - m]);
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

c(n)=Product[A029826(i),{i,0,n)];

t(n,m)=c(n)-c(m)-c(n-m)

A172986 a(0) = 0, a(n) = A029826(n+1) for n <= 20, otherwise a(n) = a(n - 1 - n % 20) + A029826(2 + n % 20 ).

Original entry on oeis.org

0, 1, -1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 3, 2, 4, 3, 3, 5, 4, 4, 5, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 7, 6, 8, 7, 8, 6, 8, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 10, 9, 11, 10, 11, 9, 11, 10, 10, 11, 10, 11, 10, 11, 11, 11, 11, 12, 11, 12, 13, 12, 14, 13, 14, 12, 14, 13, 13, 14, 13, 14, 13

Offset: 0

Roger L. Bagula, Feb 06 2010

Cf. A029826.

p[x_] = (x^(10) + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1);
q[x_] = Expand[x^10*p[1/x]];
a = Table[SeriesCoefficient[Series[1/q[x], {x, 0, 50}], n], {n, 0, 20}];
b[0] := 0;
b[n_] := b[n] = If[n <= 20, a[[n]], b[n - 1 - Mod[n, 20]] + a[[1 + Mod[n, 20]]]];
Table[b[n], {n, 0, 100}]

A117791 Expansion of 1/(1 - x - x^2 + x^4 - x^6).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 20, 30, 45, 68, 102, 153, 230, 345, 518, 778, 1168, 1754, 2634, 3955, 5939, 8918, 13391, 20108, 30194, 45339, 68081, 102230, 153508, 230507, 346128, 519744, 780445, 1171912, 1759737, 2642412, 3967832, 5958076, 8946616, 13434192

Offset: 0

Roger L. Bagula, Apr 15 2006

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

Cf. A029826, A143419, A143438, A143472, A143619, A143644, 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-x^2+x^4-x^6))); // G. C. Greubel, Nov 03 2018

seq(coeff(series(1/(1 -x -x^2 +x^4 -x^6), x, n+1), x, n), n = 0..50); # G. C. Greubel, Dec 05 2019

CoefficientList[Series[1/(1 -x -x^2 +x^4 -x^6), {x, 0, 50}], x]

Vec(1/(1 -x -x^2 +x^4 -x^6)+O(x^50)) \\ Charles R Greathouse IV, Sep 23 2012

def A117791_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1 -x -x^2 +x^4 -x^6) ).list()
A117791_list(50) # G. C. Greubel, Dec 05 2019

a(n) = a(n-1) + a(n-2) - a(n-4) + a(n-6). - Ilya Gutkovskiy, Nov 16 2016

Edited by N. J. A. Sloane, Nov 08 2006

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

cp's OEIS Frontend

A172497 Triangle T(n, k) = round( c(n)/(c(k)*c(n-k)) ) where c(n) = Product_{j=1..n} A029826(j+10), read by rows.

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A173894 a(n) = ceiling(A029826(n)/2).

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A172972 Subtraction triangle based on A029826: c(n)=Product[A029826(i),{i,0,n)];t(n,m)=c(n)-c(m)-c(n-m).

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A172986 a(0) = 0, a(n) = A029826(n+1) for n <= 20, otherwise a(n) = a(n - 1 - n % 20) + A029826(2 + n % 20 ).

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A117791 Expansion of 1/(1 - x - x^2 + x^4 - x^6).

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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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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.