|
Search: id:A058200
|
|
|
| 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. |
|
+0
1
|
|
| 9, 9, 3888, 1679616, 725594112, 313456656384, 135413275557888, 58498535041007616, 25271367137715290112, 10917230603493005328384, 4716243620708978301861888, 2037417244146278626404335616
(list; graph; listen)
|
|
|
OFFSET
|
0,1
|
|
|
COMMENT
|
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).
|
|
FORMULA
|
a(n) = 9*432^(n-1) for n>0.
|
|
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]]
|
|
CROSSREFS
|
Sequence in context: A065242 A050720 A124116 this_sequence A067450 A109409 A160563
Adjacent sequences: A058197 A058198 A058199 this_sequence A058201 A058202 A058203
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
Yasutoshi Kohmoto (zbi74583(AT)boat.zero.ad.jp)
|
|
EXTENSIONS
|
Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 06 2002
|
|
|
Search completed in 0.002 seconds
|
