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.
%I A078712 #87 Oct 01 2024 20:13:01
%S A078712 -3,1,-1,-2,3,-4,2,1,-5,7,-6,1,6,-12,13,-7,-5,18,-25,20,-2,-23,43,-45,
%T A078712 22,21,-66,88,-67,1,87,-154,155,-68,-86,241,-309,223,18,-327,550,-532,
%U A078712 205,345,-877,1082,-737,-140,1222,-1959,1819,-597,-1362
%N A078712 Series expansion of (-3 - 2*x)/(1 + x - x^3) in powers of x.
%C A078712 This sequence is -A001608(-n), the Perrin sequence for negative n. - _T. D. Noe_, Oct 10 2006
%C A078712 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
%H A078712 Vincenzo Librandi, <a href="/A078712/b078712.txt">Table of n, a(n) for n = 0..1000</a>
%H A078712 Robert Dougherty-Bliss, <a href="https://arxiv.org/abs/2206.14852">The Meta-C-finite Ansatz</a>, arXiv:2206.14852 [math.CO], 2022. See page 7.
%H A078712 Robert Dougherty-Bliss, <a href="https://sites.math.rutgers.edu/~zeilberg/Theses/RobertDoughertyBlissThesis.pdf">Experimental Methods in Number Theory and Combinatorics</a>, Ph. D. Dissertation, Rutgers Univ. (2024). See pp. 56, 58.
%H A078712 Yüksel Soykan, <a href="https://arxiv.org/abs/1910.03490">Summing Formulas For Generalized Tribonacci Numbers</a>, arXiv:1910.03490 [math.GM], 2019.
%H A078712 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,1).
%F A078712 a(n) = a(n-3) - a(n-1) with a(0)=-3, a(1)=1, a(2)=-1.
%F A078712 a(n) = A001945(n) - A001608(n).
%F A078712 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).
%F A078712 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
%F A078712 G.f.: (2*x + 3)/(x^3 - x - 1). - _Vincenzo Librandi_, May 17 2013
%e A078712 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 + ...
%t A078712 CoefficientList[Series[(2x + 3)/(x^3 - x - 1), {x, 0, 60}], x] (* _Harvey P. Dale_, Mar 18 2012 *)
%t A078712 LinearRecurrence[{-1, 0, 1}, {-3, 1, -1}, 60] (* _Harvey P. Dale_, Mar 18 2012 *)
%t A078712 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 *)
%t A078712 Table[RootSum[-1 - # + #^3 &, #^(-n) &], {n, 0, 20}] (* _Eric W. Weisstein_, Jun 27 2018 *)
%t A078712 RootSum[-1 - # + #^3 &, #^-Range[0, 20] &] (* _Eric W. Weisstein_, Jun 27 2018 *)
%o A078712 (PARI) Vec((2*x+3)/(x^3-x-1)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o A078712 (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 */
%o A078712 (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
%Y A078712 Cf. A001608, A001945, A210462, A210463.
%K A078712 sign,easy
%O A078712 0,1
%A A078712 _Ralf Stephan_, Dec 19 2002
%E A078712 Deleted certain dangerous or potentially dangerous links. - _N. J. A. Sloane_, Jan 30 2021