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A058200 Coefficients of the highest power of r in a sequence of parametric solutions for the Diophantine equation x^3+y^3+z^3=1. 1
9, 9, 3888, 1679616, 725594112, 313456656384, 135413275557888, 58498535041007616, 25271367137715290112, 10917230603493005328384, 4716243620708978301861888, 2037417244146278626404335616, 880164249471192366606672986112, 380230955771555102374082730000384 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The iteration for the solutions is as follows: x_0=9*r^4, y_0=-9*r^4+3*r, z_0=-9*r^3+1, x_1=9*r^4, y_1=-9*r^4-3*r, z_1=9*r^3+1. For n>=2: x_n=(432*r^6-2) * x_(n-1) - x_(n-2) - 108*r^4, y_n=(432*r^6-2) * y_(n-1) - y_(n-2) - 108*r^4, z_n=(432*r^6-2) * z_(n-1) - z_(n-2) + 216*r^6 + 4. For all n, x_n^3+y_n^3+z_n^3=1. Each x_n, y_n and z_n is a polynomial in r; the leading coefficients of x_n and z_n are both a(n); the leading coefficient of y_n is -a(n).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (432).

FORMULA

a(n) = 9*432^(n-1) for n>0.

a(n) = 3^(3*n-1)*16^(n-1) for n>0. G.f.: 9*(431*x-1) / (432*x-1). - Colin Barker, Jun 26 2014

EXAMPLE

For n=3, x_3 = 1679616*r^16 - 66096*r^10 + 153*r^4, y_3 = -1679616*r^16 - 559872*r^13 - 27216*r^10 + 3888*r^7 + 63*r^4 - 3*r, z_3 = 1679616*r^15 + 279936*r^12 - 11664*r^9 - 648*r^6 + 9*r^3 + 1, so a(3) = 1679616.

MATHEMATICA

x[0]=x[1]=9*r^4; y[0]=-9*r^4+3*r; z[0]=-9*r^3+1; y[1]=-9*r^4-3*r; z[1]=9*r^3+1; x[n_] := x[n]=Expand[(432*r^6-2)*x[n-1]-x[n-2]-108*r^4]; y[n_] := y[n]=Expand[(432*r^6-2)*y[n-1]-y[n-2]-108*r^4]; z[n_] := z[n]=Expand[(432*r^6-2)*z[n-1]-z[n-2]+216*r^6+4]; a[n_] := Last[CoefficientList[x[n], r]]

PROG

(PARI) Vec(9*(431*x-1)/(432*x-1) + O(x^100)) \\ Colin Barker, Jun 26 2014

CROSSREFS

Sequence in context: A124116 A213154 A226050 * A067450 A220450 A318147

Adjacent sequences:  A058197 A058198 A058199 * A058201 A058202 A058203

KEYWORD

nonn,easy

AUTHOR

Yasutoshi Kohmoto

EXTENSIONS

Edited by Dean Hickerson, Dec 06 2002

More terms from Colin Barker, Jun 26 2014

STATUS

approved

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Last modified September 18 20:18 EDT 2019. Contains 327181 sequences. (Running on oeis4.)