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

1,4

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

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-6,8).

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)

Example

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A112259, A112261 (first differences).

Sequence in context: A157827 A242276 A196374 * A196114 A250468 A259515

Adjacent sequences: A112257 A112258 A112259 * A112261 A112262 A112263

Keyword

sign,easy

Author

Russell Walsmith, Aug 30 2005

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

Status

approved