|
|
A246453
|
|
Lucas numbers (A000204) of the form n^2 + 2.
|
|
7
|
|
|
3, 11, 18, 123, 843, 5778, 39603, 271443, 1860498, 12752043, 87403803, 599074578, 4106118243, 28143753123, 192900153618, 1322157322203, 9062201101803, 62113250390418, 425730551631123, 2918000611027443, 20000273725560978, 137083915467899403, 939587134549734843
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
a(n) = {11} union {A000204(2+4*n)} for n=0,1,...
|
|
LINKS
|
|
|
FORMULA
|
G.f.: x*(3 - 10*x - 56*x^2 + 8*x^3) / (1 - 7*x + x^2).
a(n) = (2^(-n)*((7+3*sqrt(5))^n*(-20+9*sqrt(5)) + (7-3*sqrt(5))^n*(20+9*sqrt(5)))) / sqrt(5) for n>2.
a(n) = 7*a(n-1) - a(n-2) for n>4. (End)
|
|
MAPLE
|
with(combinat, fibonacci):lst:={}:lst1:={}:nn:=5000:
for n from 1 to nn do:
lst:=lst union {2*fibonacci(n-1)+fibonacci(n)}:
od:
for m from 1 to nn do:
if {m^2+2} intersect lst = {m^2+2}
then
lst1:=lst1 union {m^2+2}:
else
fi:
od:
print(lst1):
|
|
MATHEMATICA
|
CoefficientList[Series[x*(3-10*x-56*x^2+8*x^3)/(1-7*x+x^2), {x, 0, 50}], x] (* or *) LinearRecurrence[{7, -1}, {3, 11, 18, 123}, 30] (* G. C. Greubel, Dec 21 2017 *)
Select[LucasL[Range[100]], IntegerQ[Sqrt[#-2]]&] (* Harvey P. Dale, Dec 31 2018 *)
|
|
PROG
|
(PARI) lista(nn) = for (n=0, nn, luc = fibonacci(n+1) + fibonacci(n-1); if (issquare(luc-2), print1(luc, ", "))); \\ Michel Marcus, Mar 29 2016
(PARI) Vec(x*(3 - 10*x - 56*x^2 + 8*x^3) / (1 - 7*x + x^2) + O(x^30)) \\ Colin Barker, Jun 20 2017
(Magma) I:=[3, 11, 18, 123]; [n le 4 select I[n] else 7*Self(n-1)-Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|