A219300 -id:A219300 - cp's OEIS frontend

A167289 Signature sequence of the smallest Salem number of degree 18 (A219300).

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 8, 1, 7, 6, 5, 4, 3, 9, 2, 8, 1, 7, 6, 5, 4, 10, 3, 9, 2, 8, 1, 7, 6, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 1, 7, 13, 6, 12, 5, 11, 4, 10, 3, 9, 2, 8, 14, 1, 7, 13, 6, 12, 5, 11, 4, 10, 3, 9, 15

Offset: 1

Roger L. Bagula, Nov 01 2009

G. C. Greubel, Table of n, a(n) for n = 1..5000
Michael Mossinghoff, Lehmer's Problem Website.
Michael Mossinghoff, Small Salem Numbers.
Eric Weisstein's World of Mathematics, Signature Sequence.
Index entries for sequences related to signature sequences

Cf. A073011, A147851, A219300.

a = {1, -1, 1, -1, 0, 0, -1, 1, -1};
b = Join[a, {1}, Reverse[a]];
p[x_] = Sum[b[[n]]*x^(n - 1), {n, 1, Length[b]}];
m = Root[p[x], 2];
Take[Transpose[Sort[Flatten[Table[{i + j*m, i}, {i, 25}, {j, 17}], 1], #1[[1]] < #2[[1]] &]][[2]], 95]

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

A306078 Decimal expansion of the third smallest known Salem number.

Original entry on oeis.org

1, 2, 0, 0, 0, 2, 6, 5, 2, 3, 9, 8, 7, 3, 9, 1, 5, 1, 8, 9, 0, 2, 9, 6, 2, 1, 0, 0, 4, 1, 4, 6, 0, 1, 5, 6, 7, 2, 4, 0, 6, 1, 8, 1, 5, 1, 9, 9, 9, 8, 5, 1, 0, 6, 7, 9, 2, 4, 3, 9, 9, 8, 3, 9, 8, 8, 6, 0, 7, 3, 1, 1, 3, 4, 4, 2, 5, 2, 4, 0, 9, 6, 4, 4, 2, 4, 6, 1, 7, 2, 7, 8, 8, 4, 9, 6, 9, 9, 1, 0

Offset: 1

Jean-François Alcover, Jun 19 2018

			1.200026523987391518902962100414601567240618151999851067924399839886...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 14.

Cf. A073011 (sigma1), A219300 (sigma2), A306079 (sigma4).

c1 = {1, 0, 0, -1, -1, 0, 0, 1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2]-1]). c2;
sigma3 = Root[p, x, 2];
RealDigits[sigma3, 10, 100][[1]]

polrootsreal(x^14 - x^11 - x^10 + x^7 - x^4 - x^3 + 1)[2] \\ Charles R Greathouse IV, Feb 11 2025

A306079 Decimal expansion of the fourth smallest known Salem number.

Original entry on oeis.org

1, 2, 0, 2, 6, 1, 6, 7, 4, 3, 6, 8, 8, 6, 0, 4, 2, 6, 1, 1, 1, 8, 2, 9, 5, 4, 1, 5, 9, 4, 8, 6, 1, 9, 0, 4, 5, 3, 4, 3, 9, 4, 9, 8, 3, 4, 9, 6, 9, 5, 2, 3, 0, 4, 3, 6, 8, 5, 3, 0, 9, 5, 7, 6, 7, 2, 6, 4, 5, 4, 0, 6, 5, 8, 7, 6, 3, 6, 5, 5, 5, 3, 7, 7, 2, 6, 7, 1, 0, 8, 0, 0, 5, 5, 1, 8, 2, 6, 5, 7, 6, 7

Offset: 1

Jean-François Alcover, Jun 19 2018

			1.20261674368860426111829541594861904534394983496952304368530957672645...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 14.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3).

c1 = {1, 0, -1, 0, 0, 0, 0, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2]-1]).c2;
sigma4 = Root[p, x, 2];
RealDigits[sigma4, 10, 102][[1]]

polrootsreal(x^14 - x^12 - x^7 - x^2 + 1)[2] \\ Charles R Greathouse IV, Feb 11 2025

A316605 Decimal expansion of the fifth smallest known Salem number.

Original entry on oeis.org

1, 2, 1, 6, 3, 9, 1, 6, 6, 1, 1, 3, 8, 2, 6, 5, 0, 9, 1, 6, 2, 6, 8, 0, 6, 3, 1, 1, 1, 9, 9, 4, 6, 3, 3, 2, 7, 7, 2, 2, 2, 5, 3, 6, 0, 6, 5, 7, 0, 5, 7, 0, 7, 5, 7, 5, 6, 0, 4, 2, 7, 0, 6, 5, 8, 3, 8, 3, 1, 2, 1, 2, 9, 4, 6, 1, 8, 4, 9, 5, 9, 4, 4, 2, 6, 3, 7, 9, 6, 6, 6, 7, 0, 9, 5, 4, 3, 9, 5, 4, 2, 8

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.21639166113826509162680631119946332772225360657057075756042706583831...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 10.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3), A306079 (sigma4), A316606 (sigma6), A316607 (sigma7), A316608 (sigma8), A316609 (sigma9), A316610 (sigma10).

c1 = {1, 0, 0, 0, -1, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma5 = Root[p, x, 2];
RealDigits[sigma5, 10, 102][[1]]

polrootsreal(1 - x^4 - x^5 - x^6 + x^10)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x^4 - x^5 - x^6 + x^10.

A316606 Decimal expansion of the sixth smallest known Salem number.

Original entry on oeis.org

1, 2, 1, 9, 7, 2, 0, 8, 5, 9, 0, 4, 0, 3, 1, 1, 8, 4, 4, 1, 6, 9, 6, 0, 6, 7, 6, 0, 4, 1, 4, 6, 7, 7, 9, 4, 4, 3, 9, 0, 4, 1, 5, 5, 0, 5, 5, 4, 1, 5, 6, 9, 6, 7, 8, 2, 8, 7, 9, 7, 4, 4, 1, 7, 8, 7, 3, 3, 8, 4, 6, 4, 5, 9, 9, 0, 8, 3, 9, 0, 6, 5, 8, 3, 5, 5, 3, 9, 3, 2, 0, 7, 8, 5, 1, 6, 2, 5, 9, 5, 7, 8

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.219720859040311844169606760414677944390415505541569678287974417873...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 18.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316607 (sigma7), A316608 (sigma8), A316609 (sigma9), A316610 (sigma10).

c1 = {1, -1, 0, 0, 0, 0, 0, 0, -1, 1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma6 = Root[p, x, 2];
RealDigits[sigma6, 10, 102][[1]]

polrootsreal(1 - x - x^8 + x^9 - x^10 - x^17 + x^18)[2] \\ Charles R Greathouse IV, Feb 11 2025

Equals root of p = 1 - x - x^8 + x^9 - x^10 - x^17 + x^18 with largest absolute value.

A316607 Decimal expansion of the seventh smallest known Salem number.

Original entry on oeis.org

1, 2, 3, 0, 3, 9, 1, 4, 3, 4, 4, 0, 7, 2, 2, 4, 7, 0, 2, 7, 9, 0, 1, 7, 7, 9, 3, 8, 9, 7, 5, 2, 7, 9, 0, 1, 7, 5, 6, 6, 5, 7, 4, 4, 8, 9, 6, 6, 1, 7, 5, 6, 2, 4, 1, 4, 0, 1, 9, 1, 4, 2, 3, 6, 1, 7, 2, 8, 1, 3, 4, 4, 7, 8, 5, 3, 5, 4, 5, 4, 1, 6, 7, 3, 5, 9, 8, 4, 6, 5, 1, 6, 6, 2, 4, 0, 8, 5, 2, 8, 5, 6

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.2303914344072247027901779389752790175665744896617562414019142361728...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 10.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316606 (sigma6), A316608 (sigma8), A316609 (sigma9), A316610 (sigma10).

c1 = {1, 0, 0, -1, 0, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma7 = Root[p, x, 2];
RealDigits[sigma7, 10, 102][[1]]

polrootsreal(1 - x^3 - x^5 - x^7 + x^10)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x^3 - x^5 - x^7 + x^10.

A316608 Decimal expansion of the eighth smallest known Salem number.

Original entry on oeis.org

1, 2, 3, 2, 6, 1, 3, 5, 4, 8, 5, 9, 3, 1, 2, 1, 0, 0, 3, 9, 6, 2, 7, 3, 1, 6, 9, 4, 8, 0, 7, 9, 0, 9, 7, 9, 1, 4, 1, 1, 5, 7, 7, 3, 7, 1, 2, 0, 9, 8, 3, 1, 0, 4, 6, 7, 2, 9, 9, 1, 6, 5, 8, 2, 0, 5, 3, 8, 3, 9, 3, 5, 1, 3, 7, 3, 8, 9, 5, 7, 0, 3, 1, 3, 4, 9, 7, 8, 4, 8, 8, 5, 3, 0, 3, 2, 4, 9, 2, 2, 3, 3

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.2326135485931210039627316948079097914115773712098310467299165820538...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 20.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316606 (sigma6), A316607 (sigma7), A316609 (sigma9), A316610 (sigma10).

c1 = {1, -1, 0, 0, 0, -1, 1, 0, 0, -1, 1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma8 = Root[p, x, 2];
RealDigits[sigma8, 10, 102][[1]]

polrootsreal(1 - x - x^5 + x^6 - x^9 + x^10 - x^11 + x^14 - x^15 - x^19 + x^20)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x - x^5 + x^6 - x^9 + x^10 - x^11 + x^14 - x^15 - x^19 + x^20.

A316609 Decimal expansion of the ninth smallest known Salem number.

Original entry on oeis.org

1, 2, 3, 5, 6, 6, 4, 5, 8, 0, 3, 8, 9, 7, 4, 7, 3, 0, 8, 1, 0, 5, 1, 6, 9, 3, 5, 1, 5, 3, 1, 2, 6, 3, 4, 7, 9, 7, 2, 3, 5, 1, 0, 0, 4, 2, 7, 4, 6, 2, 3, 9, 0, 7, 7, 6, 5, 0, 4, 3, 8, 0, 7, 7, 2, 0, 6, 3, 1, 1, 9, 2, 4, 9, 4, 0, 2, 7, 8, 1, 4, 9, 8, 8, 9, 1, 6, 6, 0, 7, 4, 0, 4, 2, 6, 3, 8, 5, 9, 2, 5

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.2356645803897473081051693515312634797235100427462390776504380772063...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 22.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316606 (sigma6), A316607 (sigma7), A316608 (sigma8), A316610 (sigma10).

c1 = {1, 0, -1, -1, 0, 0, 0, 1, 1, 0, -1, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma9 = Root[p, x, 2];
RealDigits[sigma9, 10, 101][[1]]

polrootsreal(1 - x^2 - x^3 + x^7 + x^8 - x^10 - x^11 - x^12 + x^14 + x^15 - x^19 - x^20 + x^22)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x^2 - x^3 + x^7 + x^8 - x^10 - x^11 - x^12 + x^14 + x^15 - x^19 - x^20 + x^22.

A316610 Decimal expansion of the tenth smallest known Salem number.

Original entry on oeis.org

1, 2, 3, 6, 3, 1, 7, 9, 3, 1, 8, 0, 3, 2, 3, 0, 4, 8, 9, 8, 9, 9, 0, 9, 4, 8, 6, 9, 8, 0, 2, 0, 5, 4, 5, 5, 3, 9, 4, 4, 8, 1, 9, 2, 0, 8, 3, 6, 7, 8, 6, 9, 5, 6, 3, 7, 9, 4, 7, 5, 3, 7, 8, 4, 1, 1, 1, 8, 3, 6, 9, 9, 9, 5, 6, 7, 1, 4, 1, 5, 6, 3, 4, 2, 7, 2, 4, 3, 8, 5, 4, 3, 6, 5, 4, 0, 5, 5, 6, 1, 8, 8

Offset: 1

Jean-François Alcover, Jul 08 2018

			1.2363179318032304898990948698020545539448192083678695637947537841118...

M. J. Mossinghoff, Small Salem Numbers
Eric Weisstein's MathWorld, Salem Constants.
Wikipedia, Salem number
Index entries for algebraic numbers, degree 16.

Cf. A073011 (sigma1), A219300 (sigma2), A306078 (sigma3 ), A306079 (sigma4), A316605 (sigma5), A316606 (sigma6), A316607 (sigma7), A316608 (sigma8), A316609 (sigma9).

c1 = {1, -1, 0, 0, 0, 0, 0, 0, -1};
c2 = Join[c1, Reverse[Most[c1]]];
p = (x^Range[0, Length[c2] - 1]).c2;
sigma10 = Root[p, x, 2];
RealDigits[sigma10, 10, 102][[1]]

polrootsreal(1 - x - x^8 - x^15 + x^16)[2] \\ Charles R Greathouse IV, Feb 11 2025

p = 1 - x - x^8 - x^15 + x^16.

cp's OEIS Frontend

A167289 Signature sequence of the smallest Salem number of degree 18 (A219300).

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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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A306078 Decimal expansion of the third smallest known Salem number.

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A306079 Decimal expansion of the fourth smallest known Salem number.

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A316605 Decimal expansion of the fifth smallest known Salem number.

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A316606 Decimal expansion of the sixth smallest known Salem number.

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A316607 Decimal expansion of the seventh smallest known Salem number.

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A316608 Decimal expansion of the eighth smallest known Salem number.

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