A219511 Pierce expansion of 144 - 64*sqrt(5).

1, 9, 40, 161, 648, 51841, 207368, 5374978561, 21499914248, 57780789062419261441, 231123156249677045768, 6677239169351578707225356193679818792961, 26708956677406314828901424774719275171848

Offset: 0

Author

Peter Bala, Nov 23 2012

Keywords

• nonn
• easy

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 = 9. For other cases see A219508 (p = 3), A219509 (p = 5) and A219510 (p = 7).

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

Crossrefs

Cf. A081459, A219508 (p = 3), A219509 (p = 5), A219510 (p = 7).

Programs

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[144 - 64*Sqrt[5] , 7!], 10] (* G. C. Greubel, Nov 15 2016 *)

PARI

r=(9 + 4*sqrt(5))/16; for(n=1, 10, print(floor(r), ", "); r=r/(r-floor(r))) \\ G. C. Greubel, Nov 15 2016

Formula

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

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

Recurrence equations: a(0) = 1, a(1) = 9 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.

144 - 64*sqrt(5) = 1 - 1/9 + 1/(9*40) - 1/(9*40*161) + 1/(9*40*161*648) - ....

a(2*n) = 8*A081459(n)^2 for n >= 2.

a(2*n+1) = A081459(n+2) for n >= 0.