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 A279890 Expansion of x*(1 - x + 2*x^3 - x^4)/((1 - x)*(1 + x)*(1 - x + x^2)*(1 - x - x^2)). 0
 0, 1, 1, 2, 4, 7, 12, 19, 31, 50, 82, 133, 216, 349, 565, 914, 1480, 2395, 3876, 6271, 10147, 16418, 26566, 42985, 69552, 112537, 182089, 294626, 476716, 771343, 1248060, 2019403, 3267463, 5286866, 8554330, 13841197, 22395528, 36236725, 58632253, 94868978, 153501232, 248370211, 401871444, 650241655, 1052113099, 1702354754 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The integer part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2). The o.g.f. for the numerators of the fractional part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2) is 6*x/((1 + x - x^2)*(1 - 4*x - x^2)). The o.g.f. for the denominators of the fractional part of the harmonic mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2) is (1 + 3*x - x^2)/((1 + x)*(1 - 3*x + x^2)). Convolution of Fibonacci numbers and periodic sequence [1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, ...]. LINKS Table of n, a(n) for n=0..45. Eric Weisstein's World of Mathematics, Fibonacci Number Eric Weisstein's World of Mathematics, Harmonic Mean Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,0,-1). FORMULA G.f.: x*(1 - x + 2*x^3 - x^4)/((1 - x)*(1 + x)*(1 - x + x^2)*(1 - x - x^2)). a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - a(n-6). a(n) = (9*sqrt(5)*(((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n) + 5*((-1)^n + 2*cos(Pi*n/3) - 3))/30. a(n) = floor(3*F(n)*F(n+1)*F(n+2)/(2*F(n+1)*F(n+2)-(-1)^n)), where F(n) is the n-th Fibonacci number (A000045). a(n) = floor(3*A065563(n)/A236428(n+1)). a(n) = 3*A000045(n)/2 + ((-1)^n + 2*cos(Pi*n/3) - 3)/6. a(n) ~ 3*phi^n/(2*sqrt(5)), where phi is the golden ratio (A001622). Lim_{n->infinity} a(n+1)/a(n) = phi. EXAMPLE a(1) = floor(3/(1/F(1)+1/F(2)+1/F(3))) = floor(3/(1/1+1/1+1/2)) = 1; a(2) = floor(3/(1/F(2)+1/F(3)+1/F(4))) = floor(3/(1/1+1/2+1/3)) = 1; a(3) = floor(3/(1/F(3)+1/F(4)+1/F(5))) = floor(3/(1/2+1/3+1/5)) = 2, etc. MATHEMATICA LinearRecurrence[{2, 0, -2, 2, 0, -1}, {0, 1, 1, 2, 4, 7}, 46] Table[Floor[3 Fibonacci[n] Fibonacci[n + 1] Fibonacci[n + 2]/(2 Fibonacci[n + 1] Fibonacci[n + 2] - (-1)^n)], {n, 0, 45}] PROG (PARI) concat(0, Vec((x*(1-x+2*x^3-x^4)/((1-x)*(1+x)*(1-x+x^2))) + O(x^40))) \\ Felix Fröhlich, Dec 22 2016 CROSSREFS Cf. A000045, A001622, A004695, A065563, A236428. Cf. A062114 (the integer part of the harmonic mean of Fibonacci(n+1) and Fibonacci(n+2) for n>0). Cf. A074331 (the integer part of the geometric mean of Fibonacci(n), Fibonacci(n+1) and Fibonacci(n+2)). Sequence in context: A326080 A287525 A244472 * A018147 A125892 A072642 Adjacent sequences: A279887 A279888 A279889 * A279891 A279892 A279893 KEYWORD nonn,easy AUTHOR Ilya Gutkovskiy, Dec 22 2016 STATUS approved

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Last modified May 29 08:30 EDT 2023. Contains 363029 sequences. (Running on oeis4.)