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.

Showing 1-3 of 3 results.

A073011 Decimal expansion of Lehmer's constant (also known as the Salem constant).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Aug 03 2002

Keywords

Comments

This number is algebraic of degree 10.
The Salem constant given here is the smallest known value of Mahler's measure M(f)=abs(a_d)*Product_{k=1..d}max(1,abs(b_k)) of a polynomial f(x)=Sum_{k=0..d}(a_k*x^k)=a_d*Product_{k=1..d}(x-b_k). The minimum occurs for Lehmer's polynomial (coefficients A070178) L(x)=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1 with M(L)=1.1762808... - Hugo Pfoertner, Mar 12 2006
The Salem numbers were named after the Greek mathematician Raphaël Salem (1898-1963). - Amiram Eldar, May 01 2021

Examples

			1.17628081825991750654407033847403505069341580656469...
		

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.30, p. 193.

Crossrefs

Cf. A070178 (Coefficients of Lehmer's polynomial).

Programs

  • Mathematica
    RealDigits[x/.FindRoot[x^10+x^9-Total[x^Range[3,7]]+x+1==0,{x,1,2}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Sep 08 2011 *)
    Root[ x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, 2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 05 2013 *)
  • PARI
    default(realprecision,250); L(x)=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1; solve(x=1.1,1.2,L(x))
    
  • PARI
    polrootsreal(Pol([1, 1, 0, -1, -1, -1, -1, -1, 0, 1, 1]))[2] \\ Charles R Greathouse IV, Sep 03 2014

Formula

This is the largest real root of the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.

Extensions

Edited by N. J. A. Sloane, May 01 2012

A125950 a(0)=a(1)=...=a(9)=1; a(n) = - a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) - a(n-9) - a(n-10).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 16, 18, 22, 25, 30, 35, 41, 49, 57, 67, 79, 93, 109, 129, 151, 178, 209, 246, 290, 340, 401, 471, 554, 652, 767, 902, 1061, 1248, 1468, 1727, 2031, 2390, 2810, 3306, 3889, 4574, 5381, 6329
Offset: 0

Views

Author

Luis A Restrepo (luisiii(AT)mac.com), Feb 04 2007

Keywords

Comments

a(n) = O(n^c), where c is the larger real root of x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, 1.176280818..., the smallest known Salem constant.

References

  • Wolfram, S., A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002.

Crossrefs

Programs

  • Mathematica
    LinearRecurrence[{-1,0,1,1,1,1,1,0,-1,-1},{1,1,1,1,1,1,1,1,1,1},70] (* Harvey P. Dale, May 31 2013 *)

Formula

G.f.: ( 1+2*x+2*x^2+x^3-x^5-2*x^6-3*x^7-3*x^8-2*x^9 ) / ( 1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10 ). [R. J. Mathar, Jun 30 2010]

Extensions

Edited by Don Reble, Mar 09 2007

A143335 Expansion of (1 - 2*x^3 - x^4 - 2*x^5 - x^6 - x^7 - x^8 + 2*x^9)/(1 + x - x^3 - x^4 - x^5 - x^6 - x^7 + x^9 + x^10).

Original entry on oeis.org

1, -1, 1, -2, 1, -2, 0, -1, -3, 2, -6, 1, -4, -3, -3, -5, -4, -7, -6, -9, -8, -14, -10, -18, -18, -20, -28, -27, -38, -39, -50, -57, -67, -79, -94, -109, -128, -154, -175, -213, -244, -292, -341, -400, -475, -553, -655, -768, -905, -1062, -1253, -1470, -1732
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 22 2008

Keywords

Comments

Shares the same 10th-order "Salem" linear recurrence with A029826, A173243 and A125950.

Crossrefs

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1-2*x^3-x^4 -2*x^5-x^6-x^7-x^8+2*x^9)/(1+x-x^3-x^4-x^5-x^6-x^7+x^9 +x^10) )); // G. C. Greubel, Nov 03 2018
  • Maple
    seq(coeff(series((1-2*x^3-x^4 -2*x^5-x^6-x^7-x^8+2*x^9)/(1+x-x^3-x^4-x^5-x^6-x^7 +x^9+x^10), x, n+1), x, n), n = 0..65); # G. C. Greubel, Mar 13 2020
  • Mathematica
    LinearRecurrence[{-1,0,1,1,1,1,1,0,-1,-1}, {1,-1,1,-2,1,-2,0,-1,-3,2}, 65] (* Franck Maminirina Ramaharo, Nov 02 2018 *)
  • PARI
    my(x='x+O('x^65)); Vec((1-2*x^3-x^4-2*x^5-x^6-x^7-x^8+2*x^9)/(1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10)) \\ G. C. Greubel, Nov 03 2018
    

Formula

a(n) = -a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) - a(n-9) - a(n-10). - Franck Maminirina Ramaharo, Nov 02 2018

Extensions

Edited by Assoc. Eds. of the OEIS - Jun 30 2010
Showing 1-3 of 3 results.