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 A081068 a(n) = (Lucas(4*n+2) + 2)/5, or Fibonacci(2*n+1)^2, or A081067(n)/5. 10
 1, 4, 25, 169, 1156, 7921, 54289, 372100, 2550409, 17480761, 119814916, 821223649, 5628750625, 38580030724, 264431464441, 1812440220361, 12422650078084, 85146110326225, 583600122205489, 4000054745112196, 27416783093579881 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS First differences of A103433. REFERENCES Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75. A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 19. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876 (See Corollary 1 (vii)). Giovanni Lucca, Integer Sequences and Circle Chains Inside a Circular Segment, Forum Geometricorum, Vol. 18 (2018), 47-55. Index entries for linear recurrences with constant coefficients, signature (8,-8,1). FORMULA Equals A001519(n)^2 and A058038 - 1. a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3). a(n) = (2/5)+(3/10)*{[(7/2)-(3/2)*sqrt(5)]^n+[(7/2)+(3/2)*sqrt(5)]^n}+(1/10)*sqrt(5)*{[(7/2) +(3/2)*sqrt(5)]^n-[(7/2)-(3/2)*sqrt(5)]^n}, with n>=0. - Paolo P. Lava, Dec 01 2008 a(n) = Fibonacci(2*n)*Fibonacci(2*n+2) +1. - Gary Detlefs, Apr 01 2012 G.f.: (1-4*x+x^2)/((1-x)*(x^2-7*x+1)). - Colin Barker, Jun 26 2012 Sum {n >= 0} 1/(a(n) + 1) = 1/3*sqrt(5). - Peter Bala, Nov 30 2013 MAPLE luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 40 do printf(`%d, `, (luc(4*n+2)+2)/5) od: # James A. Sellers, Mar 05 2003 MATHEMATICA CoefficientList[Series[-(1-4*x+x^2)/((x-1)*(x^2-7*x+1)), {x, 0, 40}], x] (* or *) LinearRecurrence[{8, -8, 1}, {1, 4, 25}, 50] (* Vincenzo Librandi, Jun 26 2012 *) Table[(LucasL[4*n+2] + 2)/5, {n, 0, 30}] (* G. C. Greubel, Dec 17 2017 *) PROG (MAGMA)  I:=[1, 4, 25]; [n le 3 select I[n] else 8*Self(n-1)-8*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 26 2012 (PARI) main(size)={ return(concat([1], vector(size, n, fibonacci(2*n+1)^2))) } /* Anders HellstrÃ¶m, Jul 11 2015 */ (MAGMA) [(Lucas(4*n+2) + 2)/5: n in [0..30]]; // G. C. Greubel, Dec 17 2017 (PARI) for(n=0, 30, print1(fibonacci(2*n+1)^2, ", ")) \\ G. C. Greubel, Dec 17 2017 CROSSREFS Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A081067. Sequence in context: A006880 A227693 A175255 * A163072 A278689 A140177 Adjacent sequences:  A081065 A081066 A081067 * A081069 A081070 A081071 KEYWORD nonn,easy AUTHOR R. K. Guy, Mar 04 2003 EXTENSIONS More terms from James A. Sellers, Mar 05 2003 STATUS approved

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Last modified April 24 20:29 EDT 2019. Contains 322446 sequences. (Running on oeis4.)