

A006464


Continued fraction for Sum_{n>=0} 1/4^(2^n).
(Formerly M2512)


5



0, 3, 6, 4, 4, 2, 4, 6, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 4, 2, 4, 6, 2, 4, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 4, 2, 4, 6, 4, 2, 6, 4, 2, 4, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 2, 4, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 4, 2, 4, 6, 4, 2, 6, 4, 2, 4, 4, 6, 4, 2, 6, 4, 4, 2, 4, 6, 2, 4, 6, 4, 2, 4, 4, 6, 2, 4, 6, 4, 4, 2, 4, 6, 4
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OFFSET

0,2


COMMENTS

a(n)=A004200(n) if n=0; A004200(n)+1 if n>0 (according to case u=3, b=1 of Theorem 5 (of the reference) which states that: if B(u,infinity) = Sum_{n>=0} 1/u^(2^n) = [a0, a1, a2, ...] then B(u + b,infinity) = [a0, a1+b, a2+b, a3+b,... ] (u >= 3, b >= 0)).
The sum is equal to 0.316421509021893143708079...= A078585.
After computing the first 10^5 terms and dropping the first two (0 & 3), only the numbers 2, 4 & 6 occur. Further I found no two 0's in a row and no three 2's or three 1's in a row.  Robert G. Wilson v, Dec 01 2002


REFERENCES

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..20000
J. Shallit, Letter to N. J. A. Sloane with attachment, Aug. 1979
Jeffrey Shallit, Simple continued fractions for some irrational numbers. J. Number Theory 11 (1979), no. 2, 209217 [DOI]


EXAMPLE

0.316421509021893143708079737... = 0 + 1/(3 + 1/(6 + 1/(4 + 1/(4 + ...)))).  Harry J. Smith, May 11 2009


MAPLE

u := 4: v := 7: Buv := [u, 1, [0, u1, u+1]]: for k from 2 to v do n := nops(Buv[3]): Buv := [u, Buv[2]+1, [seq(Buv[3][i], i=1..n1), Buv[3][n]+1, Buv[3][n]1, seq(Buv[3][ni], i=1..n2)]] od:seq(Buv[3][i], i=1..2^v); # first 2^v terms of A006464, Antonio G. Astudillo (aft_astudillo(AT)hotmail.com), Dec 02 2002


MATHEMATICA

ContinuedFraction[ N[ Sum[1/4^(2^n), {n, 0, Infinity}], 1000]]


PROG

(PARI) { allocatemem(932245000); default(realprecision, 25000); x=suminf(n=0, 1/4^(2^n)); x=contfrac(x); for (n=1, 20001, write("b006464.txt", n1, " ", x[n])); } \\ Harry J. Smith, May 11 2009


CROSSREFS

Sequence in context: A105559 A090038 A308291 * A233825 A159354 A196500
Adjacent sequences: A006461 A006462 A006463 * A006465 A006466 A006467


KEYWORD

nonn,cofr


AUTHOR

N. J. A. Sloane


EXTENSIONS

Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 19 2001


STATUS

approved



