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%I #47 Oct 27 2023 10:21:11
%S 1,22,7,42,33,4,38,30,50,15,43,13,56,48,24,41,0,48,22,40,39,37,23,53,
%T 55,57,45,40,5,46,50,57,28,45,46,34,2,6,7,15,25,25,13,10,59,30,13,14,
%U 7,6,15,46,23,53,59,32,24,20,11,48,35,4,4,18,33,50,7,40,16,16,1,32,24,10,43,59,23,44,51,58,11,22,26,17
%N Coefficients in sexagesimal expansion of the positive root of x^3 + 2*x^2 + 10*x = 20, first studied by Leonardo of Pisa (Fibonacci) in 1225.
%C Leonardo of Pisa (Fibonacci) found a(0), ..., a(5) but gave a(6) as 40.
%C A159992(n)/A159993(n) = Sum_{k=0..n} a(k)/60^k = (Sum_{k=0..n} a(k)*60^(n-k))/60^n; let f(x) = x^3 + 2*x^2 + 10*x - 20, then for n > 0:
%C a(n) = Max(k: f(A159992(n-1)/A159993(n - 1) + k/60^n)) < 0),
%C a(n) + 1 = Min(k: f(A159992(n - 1)/A159993(n - 1) + k/60^n)) > 0);
%C A159994(n)/A159995(n) = f(A159992(n)/A159993(n)).
%D Cox, David A., Galois theory. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2004. xx+559 pp. ISBN: 0-471-43419-1 MR2119052 (2006a:12001). See page 9.
%D A. F. Horadam, Eight hundred years young, The Australian Mathematics Teacher 31 (1975) 123-134.
%D Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers. New York: Prometheus Books (2007): 21.
%H Vincenzo Librandi, <a href="/A159990/b159990.txt">Table of n, a(n) for n = 0..5000</a>
%H Ezra Brown and Jason C. Brunson, <a href="http://www.math.vt.edu/people/brown/doc/fibo_number.pdf">Fibonacci's forgotten number</a>
%H Stanislaw Glushkov, <a href="http://dx.doi.org/10.1016/0315-0860(76)90098-7">On approximation methods of Leonardo Fibonacci</a>, Historia Mathematica 3 (1976), pp. 291-296.
%H J. J. O'Connor and E. F. Robertson, <a href="http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Fibonacci.html">Fibonacci</a>
%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/bstud/fibo.html">Fibonacci [containing the Horadam article]</a>
%H Trond Steihaug, <a href="https://arxiv.org/abs/2211.00504">Fibonacci and digit-by-digit computation; An example of reverse engineering in computational mathematics</a>, arXiv:2211.00504 [math.HO], 2022.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%e The root is 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 38/60^6 + 30/60^7 + 50/60^8 + ...
%e Leonardo's approximation 1;22.7.42.33.4.40 is to be read as 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6 = A159992(5)/A159993(5) + 40/60^6 = 1596577777 / 1166400000 ~= 1.3688081078532235 and f(1596577777/1166400000) ~= +6.7193226361369/10^10; compare this to A159992(6)/A159993(6) = A159992(5)/A159993(5) + 38/60^6 = 31931555539 / 23328000000 ~= 1.3688081078103566 and f(31931555539/23328000000) ~= -2.3239469709985/10^10.
%e Assuming that Leonardo did similar calculations, the question may arise: why he didn't find a(6) = 38 instead of 40? Supposedly he just avoided the effort to calculate f(A159992(5)/A159993(5) + k/60^6) for k = 37, 38, or 39: 37/60^6 = 37/46656000000, 38/60^6 = 19/23328000000, or 39/60^6 = 13/15552000000; finally, he did calculate only f(A159992(5)/A159993(5) + k/60^6) for k = 36 and k = 40, the less complex cases concerning sexagesimal fractional arithmetic with 36/60^6 = 1/1296000000 and 40/60^6 = 1/1166400000: f(A159992(5)/A159993(5) + 36/60^6) ~= -1.9999999988632783, f(A159992(5)/A159993(5) + 40/60^6) ~= +0.0000000006719323.
%e The latter result looks precise enough and could explain and justify Leonardo's 'rounding'.
%t RealDigits[ Solve[x^3 + 2 x^2 + 10 x - 20 == 0, x][[3, 1, 2]], 60, 111][[1]] (* _Robert G. Wilson v_, May 11 2012 *)
%o (PARI) polrootsreal(x^3+2*x^2+10*x-20)[1] \\ _Charles R Greathouse IV_, Apr 14 2014
%Y Cf. A159991, A202300.
%K nonn,base,cons
%O 0,2
%A _Reinhard Zumkeller_, May 01 2009
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