OFFSET
0,3
COMMENTS
Floretion Algebra Multiplication Program, FAMP Code: 1tesseq[A*B] with A = + .5'i - .5'k + .5i' - .5k' - 3'jj' - .5'ij' - .5'ji' - .5'jk' - .5'kj' and B = + .5'i + .5'j + .5i' + .5j' + .5'kk' + .5'ij' + .5'ji' + .5e
REFERENCES
S. Severini, A note on two integer sequences arising from the 3-dimensional hypercube, Technical Report, Department of Computer Science, University of Bristol, Bristol, UK (October 2003).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (-1,-3,-27).
FORMULA
a(n) = (3^(n+1)/2)*(cos((n+1)*arccos(1/3)) + (-1)^(n+1) ).
a(n) = - a(n-1) - 3*a(n-2) - 27*a(n-3), a(0) = -1, a(1) = 1, a(2) = -25.
a(n) = 1/4( p^(n+1) + q^(n+1) ) + (-3)^(n+1)/2 with p = 1 + 2*sqrt(2)i and q = 1 - 2*sqrt(2)i ( i^2 = -1 ).
MATHEMATICA
CoefficientList[Series[-(1+27x^2)/((1+3x)(1-2x+9x^2)), {x, 0, 40}], x] (* or *) LinearRecurrence[{-1, -3, -27}, {-1, 1, -25}, 40] (* Harvey P. Dale, Oct 03 2014 *)
PROG
(PARI) my(x='x+O('x^40)); Vec(-(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2))) \\ G. C. Greubel, Feb 19 2019
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( -(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2)) )); // G. C. Greubel, Feb 19 2019
(SageMath) (-(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 19 2019
(GAP) a:=[-1, 1, -25];; for n in [4..40] do a[n]:=-a[n-1]-3*a[n-2] - 27*a[n-3]; od; a; # G. C. Greubel, Feb 19 2019
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Creighton Dement, May 11 2005
EXTENSIONS
Edited by Ralf Stephan, Apr 09 2009
Definition corrected by Harvey P. Dale, Oct 03 2014
STATUS
approved