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 A081018 a(n) = (Lucas(4n+1)-1)/5, or Fibonacci(2n)*Fibonacci(2n+1), or A081017(n)/5. 8
 0, 2, 15, 104, 714, 4895, 33552, 229970, 1576239, 10803704, 74049690, 507544127, 3478759200, 23843770274, 163427632719, 1120149658760, 7677619978602, 52623190191455, 360684711361584, 2472169789339634, 16944503814015855, 116139356908771352, 796030994547383610 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Another interpretation of this sequence is: nonnegative integers k such that (k + 1)^2 + (2k)^2 is a perfect square. So apart from a(0) = 0, a(n) + 1 and 2a(n) form the legs of a Pythagorean triple. - Nick Hobson, Jan 13 2007 Also solution y of Diophantine equation x^2 + 4*y^2 = k^2 for which x=y+1. - Carmine Suriano, Jun 23 2010 Also the index of the first of two consecutive heptagonal numbers whose sum is equal to the sum of two consecutive triangular numbers. - Colin Barker, Dec 20 2014 Nonnegative integers k such that G(x) = k for some rational number x where G(x) = x/(1-x-x^2) is the generating function of the Fibonacci numbers. - Tom Edgar, Aug 24 2015 The integer solutions of the equation a(b+1) = (a-b)(a-b-1) or, equivalently, binomial(a, b) = binomial(a-1, b+1) are given by (a, b) = (a(n+1), A003482(n)=Fibonacci(2*n) * Fibonacci(2*n+3)) (Lind and Singmaster). - Tomohiro Yamada, May 30 2018 REFERENCES A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 28. Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Dae S. Hong, When is the Generating Function of the Fibonacci Numbers an Integer?, The College Mathematics Journal, Vol. 46, No. 2 (March 2015), pp. 110-112 D. A. Lind, The quadratic field Q(sqrt(5)) and a certain diophantine equation, Fibonacci Quart. 6(3) (1968), 86-93. David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quart. 13 (1973), 295-298. 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). a(n) = Fibonacci(3) + Fibonacci(7) + Fibonacci(11) + ... + Fibonacci(4n+3). a(n) = -(1/5) + (1/10)*(7/2-(3/2)*sqrt(5))^n - (1/10)*(7/2-(3/2)*sqrt(5))^n*sqrt(5) + (1/10)*sqrt(5)*(7/2 +(3/2)*sqrt(5))^n + (1/10)*(7/2+(3/2)*sqrt(5))^n. - Paolo P. Lava, Oct 06 2008 G.f.: x*(2-x)/((1-x)*(1-7*x+x^2)). - Colin Barker, Mar 30 2012 E.g.f.: (1/5)^(3/2)*((1+phi^2)*exp(phi^4*x) - (1 + (1/phi^2))*exp(x/phi^4) - sqrt(5)*exp(x)), where 2*phi = 1 + sqrt(5). - G. C. Greubel, Aug 24 2015 From - Michael Somos, Aug 27 2015: (Start) a(n) = -A081016(-1-n) for all n in Z. 0 = a(n) - 7*a(n+1) + a(n+2) - 1 for all n in Z. 0 = a(n)*a(n+2) - a(n+1)^2 + a(n+1) + 2 for all n in Z. 0 = a(n)*(a(n) -7*a(n+1) -1) + a(n+1)*(a(n+1) - 1) - 2 for all n in Z. (End) EXAMPLE G.f. = 2*x + 15*x^2 + 104*x^3 + 714*x^4 + 4895*x^5 + 33552*x^6 + ... 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+1)-1)/5) od: # James A. Sellers, Mar 03 2003 MATHEMATICA (LucasL[4*Range[0, 30]+1]-1)/5 (* or *) LinearRecurrence[{8, -8, 1}, {0, 2, 15}, 30] (* G. C. Greubel, Aug 24 2015, modified Jul 14 2019 *) PROG (PARI) concat(0, Vec(x*(2-x)/((1-x)*(1-7*x+x^2)) + O(x^30))) \\ Colin Barker, Dec 20 2014 (MAGMA) [(Lucas(4*n+1)-1)/5: n in [0..30]]; // Vincenzo Librandi, Aug 24 2015 (Sage) [(lucas_number2(4*n+1, 1, -1) -1)/5 for n in (0..30)] # G. C. Greubel, Jul 14 2019 (GAP) List([0..30], n-> (Lucas(1, -1, 4*n+1) -1)/5); # G. C. Greubel, Jul 14 2019 CROSSREFS Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A081017. Partial sums of A033891. Bisection of A001654 and A059840. Equals A089508 + 1. Cf. A081007, A081016, A178898. Sequence in context: A027080 A208347 A293045 * A006675 A215643 A295268 Adjacent sequences:  A081015 A081016 A081017 * A081019 A081020 A081021 KEYWORD nonn,easy AUTHOR R. K. Guy, Mar 01 2003 EXTENSIONS More terms from James A. Sellers, Mar 03 2003 STATUS approved

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Last modified August 19 04:02 EDT 2019. Contains 326109 sequences. (Running on oeis4.)