login
Expansion of -x*(8*x^2-4*x+1) / ((2*x-1)*(4*x^2-x+1)).
4

%I #51 Sep 08 2022 08:45:21

%S 1,-1,-1,11,31,19,-41,11,431,899,199,-1349,1951,15539,24119,-5269,

%T -36209,115939,522919,583451,-459649,-696301,5336599,16510411,

%U 11941231,-20545981,-1202041,215199611,488443231,164515699,-715515401,773905451,7930934351

%N Expansion of -x*(8*x^2-4*x+1) / ((2*x-1)*(4*x^2-x+1)).

%C Previous name was: Let p = the golden mean = (1+sqrt(5))/2, t = the ordered triple (-1/p,1,p). Using the rules of 'triternion' multiplication, e.g., (1,2,3)*(1,2,3) = 1,2,3 + 6,2,4 + 6,9,3 = (13,13,10), t^n gives a sequence of ordered triples, one of which is an integer = the n-th term of the sequence.

%C The signs in the pattern seems to cycle through period 12. The n-th term of this sequence is a factor of the n-th term of A112259.

%C Let M = [1, 1-p, p; p, 1, 1-p; 1-p, p, 1] a 3 X 3 matrix where p = (1 + sqrt(5))/2. All the numbers on the main diagonal of M^n are equal to a(n). - _Philippe Deléham_, Sep 19 2020

%H Colin Barker, <a href="/A112260/b112260.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,8).

%F t = (-1/p, 1, p). (a, b, c)^2 = a(a, b, c) + b(c, a, b) + c(b, c, a) = (a^2+2bc, c^2+2ab, b^2+2ac). The integer term in t^n is the n-th term.

%F From _Colin Barker_, Nov 02 2014: (Start)

%F G.f.: -x*(8*x^2-4*x+1) / ((2*x-1)*(4*x^2-x+1)).

%F a(n) = 3*a(n-1)-6*a(n-2)+8*a(n-3). (End)

%e t = (-0.618...,1,1.618...); t^2 = (3.618...,1.381...,-1). Hence a(2) = -1.

%t s = {-1/GoldenRatio, 1, GoldenRatio}; trit[lst_] := Block[{a, b, c, d, e, f}, {a, b, c} = lst[[1]]; {d, e, f} = lst[[2]]; {{a, b, c}, FullSimplify[{a*d + b*f + c*e, a*e + b*d + c*f, a*f + b*e + c*d}]}]; f[n_] := Select[ Nest[trit, {s, s}, n][[2]], IntegerQ@# &][[1]]; Table[ f[n], {n, 0, 26}]

%t CoefficientList[Series[(8 x^2 - 4 x + 1)/((1 - 2 x) (4 x^2 - x + 1)), {x, 0, 70}], x] (* _Vincenzo Librandi_, Nov 02 2014 *)

%o (PARI) Vec(-x*(8*x^2-4*x+1)/((2*x-1)*(4*x^2-x+1)) + O(x^100)) \\ _Colin Barker_, Nov 02 2014

%o (Magma) I:=[1,-1,-1]; [n le 3 select I[n] else 3*Self(n-1)-6*Self(n-2)+8*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Nov 02 2014

%Y Cf. A112259, A112261 (first differences).

%K sign,easy

%O 1,4

%A _Russell Walsmith_, Aug 30 2005

%E More terms from _Robert G. Wilson v_, May 16 2006

%E New name (using g.f. by _Colin Barker_) from _Joerg Arndt_, Nov 04 2014