A219509 Pierce expansion of 40 - 16*sqrt(6).

1, 5, 24, 49, 200, 4801, 19208, 46099201, 184396808, 4250272665676801, 17001090662707208, 36129635465198759610694779187201, 144518541860795038442779116748808, 2610701117696295981568349760414651575095962187244375364404428801

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 = 5. For other cases see A219508 (p = 3), A219510 (p = 7) and A219511 (p = 9)

Links

G. C. Greubel, Table of n, a(n) for n = 0..20

J. Paradis, P. Viader, 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*{(5 + 2*sqrt(6))^(2^n) + (5 - 2*sqrt(6))^(2^n) + 2} for n >= 0.

a(2*n+1) = 1/2*{(5 + 2*sqrt(6))^(2^n) + (5 - 2*sqrt(6))^(2^n)} for n >= 0.

Recurrence equations: a(0) = 1, a(1) = 5 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.

40 - 16*sqrt(6) = sum {n >= 0} 1/product {k = 0..n} a(k) = 1 - 1/5 + 1/(5*24) - 1/(5*24*49) + 1/(5*24*49*200) - ....

a(2*n) = 8*A084765(n-1)^2 for n >= 2.

a(2*n+1) = A084765(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[4*(10 - 4*Sqrt[6]) , 7!], 10] (* G. C. Greubel, Nov 14 2016 *)

Prog

(PARI) r=(5 + 2*sqrt(6))/8; for(n=1, 10, print(floor(r), ", "); r=r/(r-floor(r))) \\ G. C. Greubel, Nov 15 2016

Crossrefs

Cf. A084765, A219508 (p = 3), A219510 (p = 7), A219511 (p = 9).

Sequence in context: A030766 A063143 A006145 * A202326 A085646 A121546

Adjacent sequences: A219506 A219507 A219508 * A219510 A219511 A219512

Keyword

nonn,easy

Author

Peter Bala, Nov 23 2012

Status

approved