A006464 Continued fraction for Sum_{n>=0} 1/4^(2^n). (Formerly M2512)

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

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, 209-217 [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, u-1, u+1]]: for k from 2 to v do n := nops(Buv[3]): Buv := [u, Buv[2]+1, [seq(Buv[3][i], i=1..n-1), Buv[3][n]+1, Buv[3][n]-1, seq(Buv[3][n-i], i=1..n-2)]] 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", n-1, " ", x[n])); } \\ Harry J. Smith, May 11 2009

Crossrefs

Sequence in context: A105559 A090038 A308291 * A233825 A351124 A159354

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