A001351 -id:A001351 - cp's OEIS frontend

A001945 a(n+6) = -a(n+5) + a(n+4) + 3a(n+3) + a(n+2) - a(n+1) - a(n). a(n) = sign(n) if abs(n)<=3.

0, 1, 1, 1, 5, 1, 7, 8, 5, 19, 11, 23, 35, 27, 64, 61, 85, 137, 133, 229, 275, 344, 529, 599, 875, 1151, 1431, 2071, 2560, 3481, 4697, 5953, 8245, 10649, 14111, 19048, 24605, 33227, 43739, 57591, 77275, 101107, 134848, 178709, 235405, 314089, 413909

Offset: 0

N. J. A. Sloane

It seems likely that this sequence contains infinitely many primes. In the paper by Einsiedler, Everest, Ward the heuristics for the Mersenne sequence are adapted to argue that approximately c*log(N) of the first N terms should be prime, where c is constant. Numerical evidence is provided to support this. - Graham Everest (g.everest(AT)uea.ac.uk), Mar 01 2001
For n>=4 a(n-4) is the resultant of the polynomials x^3-x-1 and x^(n+1)-x^n-1. For n=4 in fact the result is 0 as we see from the identity x^5-x^4-1=(x^3-x-1)(x^2-x+1). The characteristic polynomial of the sequence is x^6+x^5-x^4-3x^3-x^2+x+1 = (x^3-x-1)*(x^3+x^2-1). - Richard Choulet, Aug 14 2007
From Peter Bala, Sep 15 2019: (Start)
This is a linear divisibility sequence of order 6. It is a particular case of a family of divisibility sequences studied by Roettger et al. The o.g.f. has the form x*d/dx(f(x)/(x^3*f(1/x))) where f(x) = x^3 - x - 1.
More generally, if f(x) = 1 + P*x + Q*x^2 + x^3 or f(x) = -1 + P*x + Q*x^2 + x^3, where P and Q are integers, then the rational function x*d/dx(f(x)/(x^3*f(1/x))) is the generating function for a linear divisibility sequence of order 6. Cf. A001351. There are corresponding results when f(x) is a monic quartic polynomial with constant term 1. (End)
Resultant of the (s_3, s_3+n) pair where s_n(X) is X^n-X-1. See Rush link. - Michel Marcus, Sep 30 2019

			G.f. = x + x^2 + x^3 + 5*x^4 + x^5 + 7*x^6 + 8*x^7 + 5*x^8 + 19*x^9 + ...

G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.
M. Hall, A slowly increasing arithmetic sequence, J. London Math. Soc., 8 (1933), 162-166.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Peter Bala, Some linear divisibility sequences of order 6
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, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
E. L. Roettger, H. C. Williams, R. K. Guy, Some extensions of the Lucas functions, Number Theory and Related Fields: In Memory of Alf van der Poorten, Series: Springer Proceedings in Mathematics & Statistics, Vol. 43, J. Borwein, I. Shparlinski, W. Zudilin (Eds.) 2013.
David E. Rush, Degree n Relatives of the Golden Ratio and Resultants of the Corresponding Polynomials, Fib. Q. 50(4), 2012, 313-325. See p. 319.
Index entries for linear recurrences with constant coefficients, signature (-1,1,3,1,-1,-1).

Cf. A001608, A078712, A104499, A001351.

import Data.List (zipWith6)
a001945 n = a001945_list !! n
a001945_list = 0 : 1 : 1 : 1 : 5 : 1 : zipWith6
   (\u v w x y z -> - u + v + 3*w + x - y - z)
     (drop 5 a001945_list) (drop 4 a001945_list) (drop 3 a001945_list)
     (drop 2 a001945_list) (drop 1 a001945_list) (drop 0 a001945_list)
-- Reinhard Zumkeller, Jan 11 2012

A001945:=z*(1+2*z+z**2+2*z**3+z**4)/(z**3-z-1)/(z**3+z**2-1); # conjectured by Simon Plouffe in his 1992 dissertation

a[0] = 0; a[1] = a[2] = a[3] = a[5] = 1; a[4] = 5; a[n_] := a[n] = -a[n - 1] + a[n - 2] + 3a[n - 3] + a[n - 4] - a[n - 5] - a[n - 6]; Table[ a[n], {n, 0, 46}] (* Robert G. Wilson v, Mar 10 2005 *)
LinearRecurrence[{-1, 1, 3, 1, -1, -1}, {0, 1, 1, 1, 5, 1}, 50] (* T. D. Noe, Jan 11 2012 *)
a[ n_] := Sign[n] SeriesCoefficient[ x * (1 + 2 x + x^2 + 2 x^3 + x^4) / (1 + x - x^2 - 3 x^3 - x^4 + x^5 + x^6), {x, 0, Abs @ n}]; (* Michael Somos, Apr 25 2014 *)

{a(n) = sign(n) * polcoeff( x * (1 + 2*x + x^2 + 2*x^3 + x^4) / (1 + x - x^2 - 3*x^3 - x^4 + x^5 + x^6) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, Apr 25 2014 */

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,-1,1,3,1,-1]^n*[0;1;1;1;5;1])[1,1] \\ Charles R Greathouse IV, Jul 19 2016

L3(n) = polsym(x^3-x-1, n)[n+1]; \\ A001608
a(n) = my(L3n=L3(n)); L3n - matdet([L3n, L3(2*n);1, L3n])/2; \\ Michel Marcus, Sep 30 2019

G.f.: (x^5+2x^4+x^3+2x^2+x)/(x^6+x^5-x^4-3x^3-x^2+x+1). - Ralf Stephan, Dec 15 2002
a(n) ~ r1^n-2*real(r2^n), with r1=1.324717957 the inverse real root of x^3+x^2-1=0 and r2=(0.87744+0.7448617i) one inverse complex root of x^3-x-1=0. With n>9, a(n) = round(r1^n-2*real(r2^n)). - Ralf Stephan, Dec 17 2002
a(n) = A001608(n) + A078712(n). - Ralf Stephan, Dec 27 2002
a(A104499(n+1)) = A204138(n). - Reinhard Zumkeller, Jan 11 2012
a(-n) = -a(n). - Michael Somos, Apr 25 2014
a(n) = (alpha^n - 1)*(beta^n - 1)*(gamma^n - 1) where alpha, beta and gamma are the zeros of x^3 - x - 1. - Peter Bala, Sep 15 2019

More terms from James Sellers, Dec 23 1999

A327541 A linear divisibility sequence of order 8.

Original entry on oeis.org

1, 4, 16, 72, 271, 1024, 3809, 13968, 50992, 185364, 672079, 2433024, 8798609, 31797532, 114864976, 414821664, 1497814207, 5407599616, 19521754897, 70471314792, 254385558128, 918256161068, 3314588702303, 11964424716288, 43186863073121

Offset: 1

Peter Bala, Sep 22 2019

Let f(x) = 1 + P*x + Q*x^2 + R*x^3 + x^4 be a monic quartic polynomial with integer coefficients. Let g(x) = x^4*f(1/x) = 1 + R*x + Q*x^2 + P*x^3 + x^4 denote the reciprocal polynomial of f(x). Then the rational function x*d/dx( log(f(x)/g(x)) ) is the generating function of a divisibility sequence satisfying a linear recurrence equation of order 8. Here we take f(x) = 1 - x - 2*x^2 - 3*x^3 + x^4 (and normalize the resulting divisibility sequence by removing a common factor of 2 from the terms of the sequence).
Roettger et al. constructed a 5-parameter family U_n(P1,P2,P3,P4,Q) of linear divisibility sequences of order 8. This sequence is the particular case of their result with parameters P1 = 4, P2 = -5, P3 = -16, P4 = 20 and Q = 1.
There are corresponding results for certain cubic polynomials - see A001945. See also A327542.

P. Bala, Some linear divisibility sequences of order 8
E. L. Roettger, H. C. Williams, R. K. Guy, Some extensions of the Lucas functions, Number Theory and Related Fields: In Memory of Alf van der Poorten, Series: Springer Proceedings in Mathematics & Statistics 43, 271-311 (2013), chapter 5.
Index entries for linear recurrences with constant coefficients, signature (4,1,-4,-16,-4,1,4,-1).

Cf. A001351, A001945, A327542.

LinearRecurrence[{4, 1, -4, -16, -4, 1, 4, -1}, {1, 4, 16, 72, 271, 1024, 3809, 13968}, 25] (* Jean-François Alcover, Nov 12 2019 *)

a(n) = (1/2) * Sum_{i = 1..4} (alpha(i)^n - 1/alpha(i)^n), where alpha(i), 1 <= i <= 4, are the zeros of the quartic polynomial 1 - x - 2*x^2 - 3*x^3 + x^4.
a(n)^2 = -(1/4) * Product_{i = 1..6} (1 - beta(i)^n), where beta(i), 1 <= i <= 6, are the zeros of the sextic polynomial x^6 + 2*x^5 + 2*x^4 - 14*x^3 + 2*x^2 + 2*x + 1.
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3) - 16*a(n-4) - 4*a(n-5) + a(n-6) + 4*a(n-7) - a(n-8).
O.g.f.: x*(x^6 - x^4 + 8*x^3 - x^2 + 1)/((x^4 - x^3 - 2*x^2 - 3*x + 1)*(x^4 - 3*x^3 - 2*x^2 - x + 1)).

A180510 G.f.: (t^5 + 2t^4 + t^3 + 2t^2 + t) / (t^6 + t^5 - 2t^4 - 5t^3 - 2*t^2 + t + 1).

Original entry on oeis.org

0, 1, 1, 2, 7, 5, 20, 27, 49, 106, 155, 331, 560, 1013, 1917, 3310, 6223, 11117, 20140, 36899, 66185, 121014, 218791, 396703, 721280, 1305025, 2371433, 4298618, 7796439, 14150029, 25652500, 46550531, 84427441, 153141122, 277824947, 503893035, 914114320, 1658100757, 3007674389, 5455918726, 9896444495, 17951959061, 32563657260

Offset: 0

N. J. A. Sloane, Jan 20 2011

An example of a sextic divisibility sequence whose characteristic polynomial has degree 6 and a 12-element dihedral Galois group. This example has a field and polynomial discriminant of 98000, which is one of the smallest possible.

			G.f. = x + x^2 + 2*x^3 + 7*x^4 + 5*x^5 + 20*x^6 + 27*x^7 + 49*x^8 + 106*x^9 + ... - _Michael Somos_, Dec 30 2022

Found by Noam D. Elkies and described in an email from Elkies to R. K. Guy, Jan 18 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300 (corrected by Ray Chandler, Jan 19 2019)
E. L. Roettger, H. C. Williams, and R. K. Guy, Some extensions of the Lucas functions, Number Theory and Related Fields: In Memory of Alf van der Poorten, Series: Springer Proceedings in Mathematics & Statistics, Vol. 43, J. Borwein, I. Shparlinski, W. Zudilin (Eds.) 2013.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (-1,2,5,2,-1,-1).

Cf. A001351, A001945, A005120, A006235.

CoefficientList[ Series[(x^5 + 2x^4 + x^3 + 2x^2 + x)/(x^6 + x^5 - 2x^4 - 5x^3 - 2x^2 + x + 1), {x, 0, 42}], x] (* Robert G. Wilson v, Jun 26 2011 *)
a[1] = 0; a[2] = 1; a[3] = 1; a[4] = 2; a[5] = 7; a[6] = 5; a[n_Integer] := a[n] = -a[n - 6] - a[n - 5] + 2 a[n - 4] + 5 a[n - 3] + 2 a[n - 2] - a[n - 1] (* Or *)
LinearRecurrence[{-1, 2, 5, 2, -1, -1}, {0, 1, 1, 2, 7, 5}, 43] (* Roger L. Bagula, Mar 16 2012 *)
a[ n_] := a[n] = Sign[n]*With[{m = Abs[n]}, If[ m<4, {0, 1, 1, 2}[[m+1]], -a[m-1] +2*a[m-2] +5*a[m-3] +2*a[m-4] -a[m-5] -a[m-6]]]; (* Michael Somos, Dec 30 2022 *)

makelist(coeff(taylor(x*(x^4+2*x^3+x^2+2*x+1)/(x^6+x^5-2*x^4-5*x^3-2*x^2+x+1), x, 0, n), x, n), n, 1, 42);  /* Bruno Berselli, Jun 05 2011 */

Vec((x^5+2*x^4+x^3+2*x^2+x)/(x^6+x^5-2*x^4-5*x^3-2*x^2+x+1)+O(x^99)) \\ Charles R Greathouse IV, Jun 06 2011

{a(n) = sign(n)*polcoeff((x^5 + 2*x^4 + x^3 + 2*x^2 + x)/(x^6 + x^5 - 2*x^4 - 5*x^3 - 2*x^2 + x + 1) + x*O(x^abs(n)), abs(n))}; /* Michael Somos, Dec 30 2022 */

a(n) = -a(-n) for all n in Z. - Michael Somos, Dec 30 2022

A327542 A linear divisibility sequence of order 8.

Original entry on oeis.org

1, 2, 16, 36, 171, 512, 2087, 6984, 26512, 92682, 341573, 1216512, 4429309, 15898766, 57595536, 207410832, 749793263, 2703799808, 9765692771, 35235657396, 127218945296, 459128080534, 1657436539337, 5982212358144, 21594204190521

Offset: 1

Peter Bala, Sep 23 2019

Let f(x) = 1 + P*x + Q*x^2 + R*x^3 + x^4 be a monic quartic polynomial with integer coefficients. Let g(x) = x^4*f(1/x) = 1 + R*x + Q*x^2 + P*x^3 + x^4 be the reciprocal polynomial of f(x). Then the rational function x*d/dx( log(f(x)/g(-x)) ) is the generating function for a divisibility sequence satisfying a linear recurrence equation of order 8. Here we take f(x) = 1 + x - 2*x^2 + 3*x^3 + x^4 (and normalize the resulting divisibility sequence by removing a common factor of 4 from the terms of the sequence).
Roettger et al. constructed a 5-parameter family U_n(P1,P2,P3,P4,Q) of linear divisibility sequences of order 8. This sequence is a particular case of their result with parameters P1 = 2, P2 = -3, P3 = 0, P4 = -16 and Q = -1.
There are corresponding results for certain cubic polynomials - see A001945. See also A327541.

P. Bala, Some linear divisibility sequences of order 8
E. L. Roettger, H. C. Williams, R. K. Guy, Some extensions of the Lucas functions, Number Theory and Related Fields: In Memory of Alf van der Poorten, Series: Springer Proceedings in Mathematics & Statistics 43, 271-311 (2013), chapter 5.
Index entries for linear recurrences with constant coefficients, signature (2,7,-6,4,6,7,-2,-1).

Cf. A001351, A001945, A327541.

a[n_] := With[{m = 1 - 2 Mod[n, 2]}, (m/4)(x^n - m/x^n) /. {Roots[1 + x - 2x^2 + 3x^3 + x^4 == 0, x] // ToRules} // Total // Round];
a /@ Range[25] (* Jean-François Alcover, Nov 11 2019 *)

a(2*n) = (1/4) * Sum_{i = 1..4} (alpha(i)^(2*n) - 1/alpha(i)^(2*n)), where alpha(i), 1 <= i <= 4, are the zeros of the quartic polynomial 1 + x - 2*x^2 + 3*x^3 + x^4.
a(2*n+1) = (-1/4) * Sum_{i = 1..4} (alpha(i)^(2*n+1) + 1/alpha(i)^(2*n+1)).
a(2*n)^2 = (-1/16) * Product_{i = 1..6} (1 - beta(i)^(2*n)), where beta(i), 1 <= i <= 6, are the zeros of the sextic polynomial x^6 + 2*x^5 + 2*x^4 - 14*x^3 + 2*x^2 + 2*x + 1.
a(2*n+1)^2 = (1/16) * Product_{i = 1..6} (1 + beta(i)^(2*n+1)).
a(n) = 2*a(n-1) + 7*a(n-2) - 6*a(n-3) + 4*a(n-4) + 6*a(n-5) + 7*a(n-6) - 2*a(n-7) - a(n-8).
O.g.f.: x*(1 + 5*x^2 - 4*x^3 - 5*x^4 - x^6)/((1 + x - 2*x^2 + 3*x^3 + x^4)*(1 - 3*x - 2*x^2 - x^3 + x^4)).

cp's OEIS Frontend

A001945 a(n+6) = -a(n+5) + a(n+4) + 3a(n+3) + a(n+2) - a(n+1) - a(n). a(n) = sign(n) if abs(n)<=3.

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A327541 A linear divisibility sequence of order 8.

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A180510 G.f.: (t^5 + 2*t^4 + t^3 + 2*t^2 + t) / (t^6 + t^5 - 2*t^4 - 5*t^3 - 2*t^2 + t + 1).

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A327542 A linear divisibility sequence of order 8.

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A180510 G.f.: (t^5 + 2t^4 + t^3 + 2t^2 + t) / (t^6 + t^5 - 2t^4 - 5t^3 - 2*t^2 + t + 1).