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 A081016 a(n) = (Lucas(4*n+3) + 1)/5, or Fibonacci(2*n+1)*Fibonacci(2*n+2), or A081015(n)/5. 14
 1, 6, 40, 273, 1870, 12816, 87841, 602070, 4126648, 28284465, 193864606, 1328767776, 9107509825, 62423800998, 427859097160, 2932589879121, 20100270056686, 137769300517680, 944284833567073, 6472224534451830 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n-1) is, together with b(n) := A089508(n), n >= 1, the solution to a binomial problem; see A089508. Numbers k such that 1 - 2*k + 5*k^2 is a square. - Artur Jasinski, Oct 26 2008 Also solution y of Diophantine equation x^2 + 4*y^2 = h^2 for which x = y-1. - Carmine Suriano, Jun 23 2010 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. 26. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1190 Index entries for linear recurrences with constant coefficients, signature (8,-8,1). FORMULA a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3). G.f.: (1 - 2*x)/((1 - x)*(1 - 7*x + x^2)). a(n) = F(1) + F(5) + F(9) +...+ F(4*n+1) = F(2*n)*F(2*n+3) + 1, where F(j) = Fibonacci(j). a(n) = (1/5) + (2/5)*(((7/2) - (3/2)*sqrt(5))^n + ((7/2) + (3/2)*sqrt(5))^n + (1/5)*sqrt(5)*(((7/2) + (3/2)*sqrt(5))^n - ((7/2) - (3/2)*sqrt(5))^n). - Paolo P. Lava, Dec 01 2008 a(n) = 7*a(n-1) - a(n-2) - 1, n >= 2. - R. J. Mathar, Nov 07 2015 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 25 do printf(`%d, `, (luc(4*n+3)+1)/5) od: # James A. Sellers, Mar 03 2003 MATHEMATICA LinearRecurrence[{8, -8, 1}, {1, 6, 40}, 30] (* Bruno Berselli, Aug 31 2017 *) PROG (PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, -8, 8]^n*[1; 6; 40])[1, 1] \\ Charles R Greathouse IV, Sep 28 2015 (PARI) first(n) = Vec((1-2*x)/((1-x)*(1-7*x+x^2)) + O(x^n)) \\ Iain Fox, Dec 19 2017 (MAGMA) [(Lucas(4*n+3) +1)/5: n in [0..30]]; // G. C. Greubel, Dec 18 2017 (Sage) [(lucas_number2(4*n+3, 1, -1) +1)/5 for n in (0..30)] # G. C. Greubel, Jul 13 2019 (GAP) List([0..30], n-> (Lucas(1, -1, 4*n+3)[2] +1)/5 ); # G. C. Greubel, Jul 13 2019 CROSSREFS Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A081015. Partial sums of A033889. Bisection of A001654. Equals A003482 + 1. Cf. A145995, A178898. Sequence in context: A289208 A244253 A123357 * A083426 A122471 A178397 Adjacent sequences:  A081013 A081014 A081015 * A081017 A081018 A081019 KEYWORD nonn,easy AUTHOR R. K. Guy, Mar 01 2003 STATUS approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)