cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A175782 Expansion of 1/(1 - x - x^20 - x^39 + x^40).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 27, 31, 36, 42, 49, 57, 66, 76, 87, 99, 112, 126, 141, 157, 174, 192, 211, 231, 254, 279, 307, 339, 376, 419, 469, 527, 594
Offset: 0

Views

Author

Roger L. Bagula, Dec 04 2010

Keywords

Comments

Limiting ratio of a(n)/a(n-1) = 1.119189829034646... .
A quasi - Salem polynomial based on the symmetrical polynomial defined by p(x,0) = 1, p(x,n) = x^(2*n) - x^(2*n - 1) - x^n - x + 1 for n>=1.
The polynomial has one real and two complex roots outside the unit circle.

Links

Crossrefs

Cf. A175739.
Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^20-x^39+x^40))); // G. C. Greubel, Nov 03 2018
  • Maple
    gf:= 1/(1-x-x^20-x^39+x^40):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 27 2012
  • Mathematica
    CoefficientList[Series[1/(1 - x - x^20 - x^39 + x^40), {x, 0, 50}], x]
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1},{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,22},70] (* Harvey P. Dale, Jun 30 2023 *)
  • PARI
    Vec(O(x^99)+1/(1 - x - x^20 - x^39 + x^40)) \\ N.B.: This yields a vector whose first component v[1] equals a(0), i.e., the offset is shifted by one. - M. F. Hasler, Dec 11 2010

Formula

a(n) = a(n-1) + a(n-20) + a(n-39) - a(n-40). - Franck Maminirina Ramaharo, Oct 31 2018

A181600 Expansion of 1/(1 - x - x^2 + x^8 - x^10).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 85, 136, 218, 349, 559, 895, 1434, 2297, 3679, 5893, 9439, 15119, 24217, 38790, 62132, 99520, 159407, 255331, 408978, 655083, 1049283, 1680695, 2692063, 4312028, 6906816, 11063033, 17720278, 28383559, 45463532, 72821479
Offset: 0

Views

Author

Roger L. Bagula, May 06 2013

Keywords

Comments

Limiting ratio is 1.60176..., the largest real root of -1 + x^2 - x^8 - x^9 + x^10. Compare this constant to Lehmer's Salem constant A073011 and the golden mean.

Links

Crossrefs

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

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 -x-x^2+x^8-x^10))); // G. C. Greubel, Nov 03 2018
  • Mathematica
    CoefficientList[Series[1/(1 - x - x^2 + x^8 - x^10), {x, 0, 50}], x]
LinearRecurrence[{1, 1, 0, 0, 0, 0, 0, -1, 0, 1}, {1, 1, 2, 3, 5, 8, 13, 21, 33, 53}, 50] (* Harvey P. Dale, Aug 11 2015 *)
  • PARI
    Vec(1/(1 -x -x^2 +x^8 -x^10) + O(x^50)) \\ G. C. Greubel, Nov 16 2016

Formula

a(n) = a(n-1) + a(n-2) - a(n-8) + a(n-10). - Franck Maminirina Ramaharo, Oct 31 2018

A219300 Decimal expansion of the second smallest known Salem number.

Original entry on oeis.org

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

Views

Author

Arkadiusz Wesolowski, Nov 17 2012

Keywords

Comments

This number is algebraic of degree 18. Sequence A073011 contains the smallest known Salem number.
It is the only root r of the polynomial x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8 + x^7 - x^6 - x^3 + x^2 - x + 1 with abs(r) > 1. - Joerg Arndt, Nov 18 2012

Examples

			1.188368147508223588142960958....

Links

Crossrefs

Cf. A073011, A173911.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8 + x^7 - x^6 - x^3 + x^2 - x + 1 == 0, {x, 1, 2}, WorkingPrecision -> 200]][[1]]
Previous Showing 21-23 of 23 results.

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