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A006267 Continued cotangent for the golden ratio.
(Formerly M3699)
18
1, 4, 76, 439204, 84722519070079276, 608130213374088941214747405817720942127490792974404 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..7

J. Shallit, Predictable regular continued cotangent expansions, J. Res. Nat. Bur. Standards Sect. B 80B (1976), no. 2, 285-290.

Eric Weisstein's World of Mathematics, Lehmer Cotangent Expansion

FORMULA

(1+sqrt(5))/2 = cot(Sum_{n>=0} (-1)^n*acot(a(n))); let b(0) = (1+sqrt(5))/2, b(n) = (b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1)) then a(n) = floor(b(n)). - Benoit Cloitre, Apr 10 2003

a(n) = A000204(3^n). - Benoit Cloitre, Sep 18 2005

a(n) = Round(c^(3^n)) where c = GoldenRatio = 1.6180339887498948482... = (sqrt(5)+1)/2. - Artur Jasinski, Sep 22 2008

Recurrence a(n+1) = a(n)^3 + 3*a(n), a(0)=4. - Artur Jasinski, Sep 24 2008

a(n+1) = Product_{k = 0..n} A002813(k). Thus a(n) divides a(n+1). - Peter Bala, Nov 22 2012

MATHEMATICA

c = N[GoldenRatio, 1000]; Table[Round[c^(3^n)], {n, 1, 8}] (* Artur Jasinski, Sep 22 2008 *)

a = {}; x = 4; Do[AppendTo[a, x]; x = x^3 + 3 x, {n, 1, 10}]; a (* Artur Jasinski, Sep 24 2008 *)

PROG

(PARI) bn=vector(100); b(n)=if(n<0, 0, bn[n]); bn[1]=(1+sqrt(5))/2; for(n=2, 10, bn[n]=(b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1)))) a(n)=floor(b(n+1))

(PARI) { default(realprecision, 10000); bn=vector(8); bn[1]=(1+sqrt(5))/2; for(n=2, 8, bn[n]=(bn[n-1]*floor(bn[n-1]) + 1)/(bn[n-1] - floor(bn[n-1]))); for (n=1, 8, write("b006267.txt", n-1, " ", floor(bn[n]))); } \\ Harry J. Smith, May 04 2009

CROSSREFS

Cf. A002666, A002667, A002668, A002813.

Sequence in context: A052271 A184272 A080989 * A273952 A201984 A210519

Adjacent sequences:  A006264 A006265 A006266 * A006268 A006269 A006270

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

The next term is too large to include.

STATUS

approved

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Last modified March 25 12:11 EDT 2019. Contains 321470 sequences. (Running on oeis4.)