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Search: id:A096641
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| A096641 |
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Decimal expansion of number with continued fraction expansion 0, 2, 4, 8, 16, ... (0 and positive powers of 2). |
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+0
1
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| 4, 4, 5, 9, 3, 4, 6, 4, 0, 5, 1, 2, 2, 0, 2, 6, 6, 8, 1, 1, 9, 5, 5, 4, 3, 4, 0, 6, 8, 2, 6, 1, 7, 6, 8, 4, 2, 7, 0, 4, 0, 8, 8, 4, 5, 2, 0, 3, 4, 3, 8, 5, 0, 7, 9, 0, 3, 2, 6, 3, 5, 6, 0, 5, 0, 0, 6, 6, 1, 9, 0, 0, 6, 9, 1, 6, 2, 3, 2, 7, 7, 8, 9, 9, 7, 7, 7, 1, 6, 1, 8, 9, 0, 3, 9, 9, 2, 1, 4, 6, 2, 0, 4, 2, 4
(list; cons; graph; listen)
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OFFSET
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0,1
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COMMENT
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According to the McLaughlin-Wyshinski paper, Tasoev proposed continued fractions of the form [a0;a,...,a,a^2,...,a^2,a^3,...,a^3,...], where a0 >= 0, a >= 2 and m >= 1 are integers and each power of a occurs m times. This sequence is for the minimal values a0 = 0, a = 2 and m = 1. Komatsu "derived a closed form for the general case (m >= 1, arbitrary)" and the expression given in (1.2) (where a0=0 and m=1) of the linked paper and which is used in the second PARI/GP program below.
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LINKS
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J. McLaughlin and N. J. Wyshinski, Ramanujan and the Regular Continued Fraction Expansion of Real Numbers, page 2.
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EXAMPLE
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0.44593464051220266811955434068261768...
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PROGRAM
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(PARI) /* Increase default precision. Stack size may need */ /* to be increased as well. */ \p 400 dec_exp(v)= w=contfracpnqn(v); w[1, 1]/w[2, 1]+0. dec_exp(vector(400, i, if(i==1, 0, 2^(i-1))) /* The following uses Komatsu's expression for given a; a0=0, m=1 */ {Komatsu(a)=suminf(s=0, a^(-(s+1)^2)*prod(i=1, s, (a^(2*i)-1)^(-1))) /suminf(s=0, a^(-s^2)*prod(i=1, s, (a^(2*i)-1)^(-1)))} Komatsu(2) /* generates this sequence's constant */
(PARI) /* Increase default precision. Stack size may need to be increased as well. */ \p 400 dec_exp(v)= w=contfracpnqn(v); w[1, 1]/w[2, 1]+0. dec_exp(vector(400, i, if(i==1, 0, 2^(i-1)))
/* The following uses Komatsu's expression for given a; a0=0, m=1 */ {Komatsu(a)=suminf(s=0, a^(-(s+1)^2)*prod(i=1, s, (a^(2*i)-1)^(-1))) /suminf(s=0, a^(-s^2)*prod(i=1, s, (a^(2*i)-1)^(-1)))} Komatsu(2) /* generates this sequence's constant */
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CROSSREFS
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Sequence in context: A058619 A095945 A072231 this_sequence A155693 A160705 A107851
Adjacent sequences: A096638 A096639 A096640 this_sequence A096642 A096643 A096644
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KEYWORD
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cons,nonn
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AUTHOR
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Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jun 30 2004
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