A112260 Expansion of -x*(8*x^2-4*x+1) / ((2*x-1)*(4*x^2-x+1)).

1, -1, -1, 11, 31, 19, -41, 11, 431, 899, 199, -1349, 1951, 15539, 24119, -5269, -36209, 115939, 522919, 583451, -459649, -696301, 5336599, 16510411, 11941231, -20545981, -1202041, 215199611, 488443231, 164515699, -715515401, 773905451, 7930934351

Offset: 1

Author

Russell Walsmith, Aug 30 2005

Keywords

• sign
• easy

Comments

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.

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.

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

Examples

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

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-6,8).

Crossrefs

Cf. A112259, A112261 (first differences).

Programs

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

Mathematica

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}]
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 *)

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

Formula

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.

From Colin Barker, Nov 02 2014: (Start)

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

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

Extensions

More terms from Robert G. Wilson v, May 16 2006

New name (using g.f. by Colin Barker) from Joerg Arndt, Nov 04 2014