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 A246453 Lucas numbers (A000204) of the form n^2 + 2. 6
 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,... 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). 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) 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 Cf. A000204 (Lucas), A059100 (n^2+2). Cf. quadrisection of A000032: A056854 (first), A056914 (second), this sequence (third, without 11), A288913 (fourth). Sequence in context: A225144 A335135 A228470 * A117769 A252802 A030377 Adjacent sequences:  A246450 A246451 A246452 * A246454 A246455 A246456 KEYWORD nonn,easy AUTHOR Michel Lagneau, Aug 26 2014 EXTENSIONS Corrected by Michel Marcus, Mar 29 2016 STATUS approved

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Last modified December 3 02:14 EST 2021. Contains 349445 sequences. (Running on oeis4.)