A001945 -id:A001945 - cp's OEIS frontend

A060169 Number of orbits of length n under the automorphism of the 3-torus whose periodic points are counted by A001945.

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 4, 4, 5, 8, 6, 12, 13, 16, 23, 26, 35, 46, 54, 76, 89, 120, 154, 192, 255, 322, 411, 544, 679, 898, 1145, 1476, 1925, 2466, 3201, 4156, 5338, 6978, 8985

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A001945 records the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(17) = 8 since the map whose periodic points are counted by A001945 has 1 fixed point and 137 points of period 17, hence 8 orbits of length 7.

Manfred Einsiedler, Graham Everest and Thomas Ward, Primes in sequences associated to polynomials (after Lehmer), LMS J. Comput. Math. 3 (2000), 125-139.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

Cf. A001642, A060164, A060165, A060166, A060167, A060168, A060170, A060171, A060171, A060172, A060173.

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001945(n/d).

A104499 Primes by index in A001945.

Original entry on oeis.org

4, 6, 8, 9, 10, 11, 15, 17, 19, 23, 25, 31, 41, 47, 59, 61, 71, 113, 127, 137, 157, 167, 223, 401, 443, 457, 503, 523, 619, 701, 859, 991, 1021, 1187, 1217, 1223, 1361, 1847, 1867, 2237, 2579, 2693, 3313, 4219, 4931, 4967, 5413, 7727, 8011, 8387, 8761, 11083, 13781, 16871, 23311, 28661, 28871, 29123

Offset: 1

Robert G. Wilson v, Mar 10 2005

nonn

All terms greater than a(11)=25 are prime so far.

a(80) > 10^6. - Paul Vanderveen, Oct 02 2022

Paul Vanderveen, Table of n, a(n) for n = 1..79

Cf. A001945.

import Data.List (findIndices)
a104499 n = a104499_list !! (n-1)
a104499_list = findIndices ((== 1) . a010051) a001945_list
-- Reinhard Zumkeller, Jan 11 2012

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]; lst = {}; Do[ If[ PrimeQ[a [ n]], AppendTo[lst, n]], {n, 21900}]; lst

A001945(a(n+1)) = A204138(n). [Reinhard Zumkeller, Jan 11 2012]

a(50) and a(52) corrected and more terms from Paul Vanderveen, Jul 27 2022

A204138 Primes in A001945, apart from initializing 6 terms.

Original entry on oeis.org

7, 5, 19, 11, 23, 61, 137, 229, 599, 1151, 5953, 101107, 549149, 16036673, 28158577, 468599149, 63088004325217, 3233514251032733, 53815701950239487, 14906561637522623977, 248091402816751952399, 1712077530466014390485776063

Offset: 1

Reinhard Zumkeller, Jan 11 2012

nonn

a(n) = A001945(A104499(n+1)).

Paul Vanderveen, Table of n, a(n) for n = 1..48

Cf. A001945.

a204138 n = a204138_list !! (n-1)
a204138_list = filter ((== 1) . a010051) $ drop 6 a001945_list

A001351 Associated Mersenne numbers.

Original entry on oeis.org

0, 1, 3, 1, 3, 11, 9, 8, 27, 37, 33, 67, 117, 131, 192, 341, 459, 613, 999, 1483, 2013, 3032, 4623, 6533, 9477, 14311, 20829, 30007, 44544, 65657, 95139, 139625, 206091, 300763, 439521, 646888, 948051, 1385429, 2033193, 2983787, 4366197, 6397723, 9387072

Offset: 0

N. J. A. Sloane, R. K. Guy

From Peter Bala, Sep 15 2019: (Start)
This is a linear divisibility sequence of order 6 (Haselgrove, p. 21). 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^2 - 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. A001945. There are corresponding results when f(x) is a monic quartic polynomial with constant term 1. (End)

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).

Danny Rorabaugh, Table of n, a(n) for n = 0..6000
Peter Bala, Some linear divisibility sequences of order 6
C. B. Haselgrove, Associated Mersenne numbers, Eureka, 11 (1949), 19-22. [Annotated and scanned copy]
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
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.
Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
Index entries for linear recurrences with constant coefficients, signature (1,-1,3,-1,1,-1).

Cf. A001350, A001945.

I:=[0,1,3,1,3,11]; [n le 6 select I[n] else Self(n-1) - Self(n-2) + 3*Self(n-3) - Self(n-4) + Self(n-5) - Self(n-6): n in [1..50]]; // Vincenzo Librandi, Sep 23 2015

A001351:=z*(z^2-z+1)*(z^2+3*z+1)/(z^3+z-1)/(z^3-z^2-1); # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{1, -1, 3, -1, 1, -1}, {0, 1, 3, 1, 3, 11}, 50] (* Vincenzo Librandi, Sep 23 2015 *)

a(n) = a(n-1) - a(n-2) + 3*a(n-3) - a(n-4) + a(n-5) - a(n-6) for n >= 6. - Sean A. Irvine, Sep 23 2015

a(n) = (alpha^n - 1)*(beta^n - 1)*(gamma^n - 1) where alpha, beta and gamma are the zeros of x^3 - x^2 - 1. - Peter Bala, Sep 15 2019

More terms from Vincenzo Librandi, Sep 23 2015

A078712 Series expansion of (-3 - 2*x)/(1 + x - x^3) in powers of x.

Original entry on oeis.org

-3, 1, -1, -2, 3, -4, 2, 1, -5, 7, -6, 1, 6, -12, 13, -7, -5, 18, -25, 20, -2, -23, 43, -45, 22, 21, -66, 88, -67, 1, 87, -154, 155, -68, -86, 241, -309, 223, 18, -327, 550, -532, 205, 345, -877, 1082, -737, -140, 1222, -1959, 1819, -597, -1362

Offset: 0

Ralf Stephan, Dec 19 2002

This sequence is -A001608(-n), the Perrin sequence for negative n. - T. D. Noe, Oct 10 2006

Similar to the Perrin sequence A001608, I conjecture that if p is a prime then a(p) == 1 (mod p). This implies that A001945(n) == 1 (mod p) and A001608(2*n) == 2 (mod p). - Michael Somos, Dec 25 2022

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Dougherty-Bliss, The Meta-C-finite Ansatz, arXiv:2206.14852 [math.CO], 2022. See page 7.
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See pp. 56, 58.
Yüksel Soykan, Summing Formulas For Generalized Tribonacci Numbers, arXiv:1910.03490 [math.GM], 2019.
Index entries for linear recurrences with constant coefficients, signature (-1,0,1).

Cf. A001608, A001945, A210462, A210463.

I:=[-3, 1, -1]; [n le 3 select I[n] else -Self(n-1)+Self(n-3): n in [1..60]]; // Vincenzo Librandi, May 17 2013

CoefficientList[Series[(2x + 3)/(x^3 - x - 1), {x, 0, 60}], x] (* Harvey P. Dale, Mar 18 2012 *)
LinearRecurrence[{-1, 0, 1}, {-3, 1, -1}, 60] (* Harvey P. Dale, Mar 18 2012 *)
a[n_] := If[n < 0, SeriesCoefficient[(-3 + x^2)/(1 - x^2 - x^3), {x, 0, -n}], SeriesCoefficient[(-3 - 2 x)/(1 + x - x^3), {x, 0, n}]]; (* Michael Somos, Oct 15 2017 *)
Table[RootSum[-1 - # + #^3 &, #^(-n) &], {n, 0, 20}] (* Eric W. Weisstein, Jun 27 2018 *)
RootSum[-1 - # + #^3 &, #^-Range[0, 20] &] (* Eric W. Weisstein, Jun 27 2018 *)

Vec((2*x+3)/(x^3-x-1)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = if( n<0, polcoeff( (-3 + x^2) / (1 - x^2 - x^3) + x * O(x^-n), -n), polcoeff( (-3 - 2*x) / (1 + x - x^3) + x * O(x^n), n))}; /* Michael Somos, Oct 15 2017 */

a(n) = a(n-3) - a(n-1) with a(0)=-3, a(1)=1, a(2)=-1.
a(n) = A001945(n) - A001608(n).
a(n) ~ 2*real(r^n) with r = 0.87743... + 0.7448617...*i one inverse complex root of x^3 - x - 1 = 0 (A210462, A210463).
2*a(n) = A001608(2*n) - A001608(n)^2 follows from the Binet formula for a(n) = -p^(-n) - r^(-n) - s^(-n), where p, r, s are roots of the Perrin polynomial x^3 - x - 1. - Roman Witula, Jan 31 2013
G.f.: (2*x + 3)/(x^3 - x - 1). - Vincenzo Librandi, May 17 2013

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A059990 Number of points of period n under the dual of the map x->2x on Z[1/6].

Original entry on oeis.org

1, 1, 7, 5, 31, 7, 127, 85, 511, 341, 2047, 455, 8191, 5461, 32767, 21845, 131071, 9709, 524287, 349525, 2097151, 1398101, 8388607, 1864135, 33554431, 22369621, 134217727, 89478485, 536870911, 119304647

Offset: 1

Thomas Ward, Mar 08 2001

This sequence counts the periodic points in the simplest nontrivial S-integer dynamical system. These dynamical systems arise naturally in arithmetic and are built by making an isometric extension of a familiar hyperbolic system. The extension destroys some of the periodic points, in this case reducing the original number 2^n-1 by factoring out any 3's. An interesting feature is that the logarithmic growth rate is still log 2.

A059990[n+7] times some power of 3 seems to me the greatest common Denominator of A035522[4n+16+1],A035522[4n+16+2],A035522[4n+16+3] and A035522[4n+16+4] for n>1 [From Dylan Hamilton, Aug 04 2010]

			a(6)=7 because 2^6-1 = 3^2x7, so |2^6-1|_3=3^(-2).

V. Chothi, G. Everest, T. Ward. S-integer dynamical systems: periodic points. J. Reine Angew. Math., 489 (1997), 99-132.
T. Ward. Almost all S-integer dynamical systems have many periodic points. Erg. Th. Dynam. Sys. 18 (1998), 471-486.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Cf. A000225, A001945, A059991.

a(n)=(2^n-1)x|2^n-1|_3

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

A060169 Number of orbits of length n under the automorphism of the 3-torus whose periodic points are counted by A001945.

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A104499 Primes by index in A001945.

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Haskell

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A204138 Primes in A001945, apart from initializing 6 terms.

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Haskell

A001351 Associated Mersenne numbers.

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A078712 Series expansion of (-3 - 2*x)/(1 + x - x^3) in powers of x.

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PARI

PARI

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A059990 Number of points of period n under the dual of the map x->2x on Z[1/6].

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A327541 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).