A087612 -id:A087612 - cp's OEIS frontend

A143335 Expansion of (1 - 2x^3 - x^4 - 2x^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).

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

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

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

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).

Cf. A029826, A070178, A087612, A107479, A107480, A109538.

Cf. A109543, A109544, A114749, A125950, A130844, A147851.

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

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

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 *)

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

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

Edited by Assoc. Eds. of the OEIS - Jun 30 2010

A059928 The periodic point counting sequence for the toral automorphism given by the polynomial of (conjectural) smallest Mahler measure. The map is x -> Ax mod 1 for x in [0,1)^10, where A is the companion matrix for the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.

Original entry on oeis.org

1, 1, 1, 9, 1, 1, 1, 9, 1, 1, 1, 81, 1, 169, 841, 9, 1, 1, 1369, 9, 1, 529, 1, 81, 2401, 625, 1, 1521, 1, 841, 1024, 8649, 4489, 1, 5041, 729, 1, 1369, 6241, 9, 6889, 169, 29929, 4761, 841, 2209, 1, 178929, 85849, 2401, 10609, 5625, 100489, 2809, 11881, 1521, 1369

Offset: 1

Graham Everest (g.everest(AT)uea.ac.uk), Mar 01 2001

nonn

It is expected that the sequence contains infinitely many squares of primes. The heuristics for the Mersenne sequence can be adapted to show that approximately c log N of the first N terms should be prime. The paper Einsiedler, Everest, Ward gives supporting numerical evidence.

The terms in this sequence are all squares. The sequence of square roots, A087612, is conjectured to contain an infinite number of primes. - T. D. Noe, Sep 15 2003

			The first term is 1 because Ax=x implies x=0 (since A-I) is invertible. Thus there is only one fixed point for the map.

G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.

Manfred Einsiedler, Graham Everest and Thomas Ward, Primes in sequences associated to polynomials (after Lehmer), LMS J. Comput. Math. 3 (2000), 125-139.
G. Everest and T. Ward, Primes in Divisibility Sequences, Cubo Matematica Educacional (2001), 3 (2), pp. 245-259.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Index to divisibility sequences

Cf. A060478, A087612.

CompanionMatrix[p_, x_] := Module[{cl=CoefficientList[p, x], deg, m}, cl=Drop[cl/Last[cl], -1]; deg=Length[cl]; If[deg==1, {-cl}, m=RotateLeft[IdentityMatrix[deg]]; m[[ -1]]=-cl; Transpose[m]]]; c=CompanionMatrix[x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, x]; im=IdentityMatrix[10]; tmp=im; Table[tmp=tmp.c; Abs[Det[tmp-im]], {n, 100}] (From T. D. Noe)

comp(pol) = my(v=Vec(pol), nn=poldegree(pol)); matrix(nn, nn, n, k, if (k==nn, -v[n], if(k==n-1, 1)));
id(nn) = matrix(nn, nn, n, k, n==k);
a(n) = my(p=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, m=comp(p)); abs(matdet(m^n-id(poldegree(p)))); \\ Michel Marcus, Nov 23 2022

The n-th term is abs(det(A^n-I)) where I is the 10 by 10 identity matrix and A is the companion matrix to the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.

More terms from T. D. Noe, Sep 15 2003

cp's OEIS Frontend

A143335 Expansion of (1 - 2x^3 - x^4 - 2x^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).

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A059928 The periodic point counting sequence for the toral automorphism given by the polynomial of (conjectural) smallest Mahler measure. The map is x -> Ax mod 1 for x in [0,1)^10, where A is the companion matrix for the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.

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cp's OEIS Frontend

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).

Views

Author

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Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A059928 The periodic point counting sequence for the toral automorphism given by the polynomial of (conjectural) smallest Mahler measure. The map is x -> Ax mod 1 for x in [0,1)^10, where A is the companion matrix for the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A143335 Expansion of (1 - 2x^3 - x^4 - 2x^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).