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

-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

0,1

Comments

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

Links

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).

Formula

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

Example

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 + ...

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A001608, A001945, A210462, A210463.

Sequence in context: A269973 A128316 A065836 * A287218 A306438 A221978

Adjacent sequences: A078709 A078710 A078711 * A078713 A078714 A078715

Keyword

sign,easy

Author

Ralf Stephan, Dec 19 2002

Extensions

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

Status

approved