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A219510
Pierce expansion of 84 - 48*sqrt(3).
4
1, 7, 32, 97, 392, 18817, 75272, 708158977, 2832635912, 1002978273411373057, 4011913093645492232, 2011930833870518011412817828051050497, 8047723335482072045651271312204201992
OFFSET
0,2
COMMENTS
Paradis et al. have determined the Pierce expansion of the quadratic irrationality 2*(p - 1)*(p - sqrt(p^2 - 1)), p a positive integer greater than or equal to 3. This is the case p = 7. For other cases see A219508 (p = 3), A219509 (p = 5) and A219511 (p = 9).
LINKS
J. Paradis, P. Viader, and L. Bibiloni Approximation to quadratic irrationals and their Pierce expansions, The Fibonacci Quarterly, Vol.36 No. 2 (1998) 146-153.
T. A. Pierce, On an algorithm and its use in approximating roots of algebraic equations, Amer. Math. Monthly, Vol. 36 No. 10, (1929) p.523-525.
Eric Weisstein's World of Mathematics, Pierce Expansion
FORMULA
a(2*n) = 2*{(2 + sqrt(3))^(2^n) + (2 - sqrt(3))^(2^n) + 2} for n >= 1.
a(2*n-1) = 1/2*{(2 + sqrt(3))^(2^n) + (2 - sqrt(3))^(2^n)} for n >= 1.
Recurrence equations: a(0) = 1, a(1) = 7 and for n >= 1 a(2*n) = 4*(a(2*n-1) + 1) and a(2*n+1) = 2*(a(2*n-1))^2 - 1.
84 - 48*sqrt(3) = 1 - 1/7 + 1/(7*32) - 1/(7*32*97) + 1/(7*32*97*392) - ....
a(2*n) = 8*A002812(n-1)^2 for n >= 1.
a(2*n+1) = A002812(n+1) for n >= 0.
MATHEMATICA
PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[84 - 48*Sqrt[3] , 7!], 10] (* G. C. Greubel, Nov 15 2016 *)
PROG
(PARI) r=(7 + 4*sqrt(3))/12; for(n=1, 10, print(floor(r), ", "); r=r/(r-floor(r))) \\ G. C. Greubel, Nov 15 2016
CROSSREFS
Cf. A002812, A219508 (p = 3), A219509 (p = 5), A219511 (p = 9).
Sequence in context: A126562 A190096 A254407 * A164270 A182820 A001794
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Nov 23 2012
STATUS
approved