OFFSET
1,1
COMMENTS
Also the real part of f(x+n*f(x,y,z), y+n*f(x,y,z), z+n*f(x,y,z))/f(x,y,z) for f(x,y,z) = x^3+y^2+z at x=(-1+i*sqrt(3))/2, y=i and z=5.
If f(x,y,z) is a trivariate polynomial, f(x+n*f(x,y,z),y+n*f(x,y,z),z+n*f(x,y,z)) is congruent to 0 (mod f(x,y,z)).
The ratio f(x+n*f,y+n*f,z+n*f)/f of these two functions is decomposed into the real part (this sequence here), and the imaginary part. The imaginary part is 2*n*i + sqrt(3)*A167469(n)*i, where i=sqrt(-1) is the imaginary unit.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(23 + 98*x + 30*x^2 - x^3)/(1-x)^4.
E.g.f.: (2 + 44*x + 145*x^2 + 50*x^3)*exp(x)/2 -1. - G. C. Greubel, Apr 09 2016
EXAMPLE
f(x +f(x,y,z), y + f(x,y,z), z + f(x,y,z)) = (23 + 2i + 6*sqrt(3)*i)* f(x,y,z) at n=1.
MAPLE
f := proc(x, y, z) x^3+y^2+z ; end proc:
A167467 := proc(n) local rho, a , x, y, z; a := f(x+n*f(x, y, z), y+n*f(x, y, z), z+n*f(x, y, z))/f(x, y, z) ; rho := (-1+I*sqrt(3))/2 ; a := subs({x = rho, y=I, z=5}, a) ; a := expand(a) ; Re(a) ; end:
seq(A167467(n), n=1..50) ; # R. J. Mathar, Nov 12 2009
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {23, 190, 652, 1559}, 50] (* Harvey P. Dale, Sep 28 2012 *)
PROG
(PARI) a(n)=1+25*n^3-n*(5*n+1)/2 \\ Charles R Greathouse IV, Jul 07 2013
(Magma) [25*n^3 - n*(5*n+1)/2 + 1: n in [1..50]]; // G. C. Greubel, Sep 01 2019
(Sage) [25*n^3 - n*(5*n+1)/2 + 1 for n in (1..50)] # G. C. Greubel, Sep 01 2019
(GAP) List([1..50], n-> 25*n^3 - n*(5*n+1)/2 + 1); # G. C. Greubel, Sep 01 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
A.K. Devaraj, Nov 05 2009
EXTENSIONS
a(2) and a(3) corrected, definition simplified and sequence extended by R. J. Mathar, Nov 12 2009
STATUS
approved