|
|
A020716
|
|
Pisot sequences E(6,8), P(6,8).
|
|
1
|
|
|
6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 113, 150, 199, 264, 350, 464, 615, 815, 1080, 1431, 1896, 2512, 3328, 4409, 5841, 7738, 10251, 13580, 17990, 23832, 31571, 41823, 55404, 73395, 97228, 128800, 170624, 226029, 299425, 396654, 525455, 696080, 922110, 1221536
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
LINKS
|
|
|
FORMULA
|
a(n) = a(n-1) + a(n-2) - a(n-4) (holds at least up to n = 1000 but is not known to hold in general).
Empirical g.f.: (6+2*x-3*x^2-4*x^3) / ((1-x)*(1-x^2-x^3)). - Colin Barker, Jun 05 2016
Theorem: E(6,8) satisfies a(n) = a(n - 1) + a(n - 2) - a(n - 4) for n>=4. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger. This shows that the above conjectures are correct. - N. J. A. Sloane, Sep 10 2016
|
|
MATHEMATICA
|
RecurrenceTable[{a[0]==6, a[1]==8, a[n]== Floor[a[n-1]^2/a[n-2] + 1/2]}, a, {n, 0, 50}] (* Bruno Berselli, Feb 05 2016 *)
|
|
PROG
|
(Magma) Exy:=[6, 8]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..50]]; // Bruno Berselli, Feb 05 2016
(PARI) Vec((6+2*x-3*x^2-4*x^3)/((1-x)*(1-x^2-x^3)) + O(x^50)) \\ Jinyuan Wang, Mar 10 2020
|
|
CROSSREFS
|
See A008776 for definitions of Pisot sequences.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|