A246453 Lucas numbers (A000204) of the form n^2 + 2.

3, 11, 18, 123, 843, 5778, 39603, 271443, 1860498, 12752043, 87403803, 599074578, 4106118243, 28143753123, 192900153618, 1322157322203, 9062201101803, 62113250390418, 425730551631123, 2918000611027443, 20000273725560978, 137083915467899403, 939587134549734843

Offset: 1

Author

Michel Lagneau, Aug 26 2014

Keywords

• nonn
• easy

Comments

a(n) = {11} union {A000204(2+4*n)} for n=0,1,...

Intersection of A000204 and A059100. - Michel Marcus, Aug 26 2014

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-1).

Crossrefs

Cf. A000204 (Lucas), A059100 (n^2+2).

Cf. quadrisection of A000032: A056854 (first), A056914 (second), this sequence (third, without 11), A288913 (fourth).

Programs

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

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 *)

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

Formula

From Colin Barker, Jun 20 2017: (Start)

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)

E.g.f.: 2*exp(7*x/2)*(9*cosh(3*sqrt(5)*x/2) - 4*sqrt(5)*sinh(3*sqrt(5)*x/2)) + 4*x^2 - 18. - Stefano Spezia, Apr 14 2025

Extensions

Corrected by Michel Marcus, Mar 29 2016