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A167467 a(n) = 25*n^3 - n*(5*n+1)/2 + 1. 2
23, 190, 652, 1559, 3061, 5308, 8450, 12637, 18019, 24746, 32968, 42835, 54497, 68104, 83806, 101753, 122095, 144982, 170564, 198991, 230413, 264980, 302842, 344149, 389051, 437698, 490240, 546827, 607609, 672736, 742358, 816625, 895687, 979694, 1068796 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A269072 A125360 A126518 * A129052 A201859 A232150
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

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Last modified April 13 12:28 EDT 2024. Contains 371641 sequences. (Running on oeis4.)