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
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