The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A123725 Numerators of fractional partial quotients appearing in a continued fraction for the power series Sum_{n>=0} x^(2^n - 1)/(n+1)^s. 3
 1, 2, -3, -2, -4, 2, 3, -2, -5, 2, -3, -2, 4, 2, 3, -2, -6, 2, -3, -2, -4, 2, 3, -2, 5, 2, -3, -2, 4, 2, 3, -2, -7, 2, -3, -2, -4, 2, 3, -2, -5, 2, -3, -2, 4, 2, 3, -2, 6, 2, -3, -2, -4, 2, 3, -2, 5, 2, -3, -2, 4, 2, 3, -2, -8, 2, -3, -2, -4, 2, 3, -2, -5, 2, -3, -2, 4, 2, 3, -2, -6, 2, -3, -2, -4, 2, 3, -2, 5, 2, -3, -2, 4, 2, 3, -2, 7, 2, -3, -2, -4, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Antti Karttunen, Table of n, a(n) for n = 0..65537 FORMULA a(n) = (A007814(n) + 2) * (-1)^A073089(n+1), n>=1, with a(0)=1. a(n) = A089080(n+1) * (-1)^A073089(n+1) for n>=0. EXAMPLE Surprisingly, the following analog of the Riemann zeta function: Z(x,s) = Sum_{n>=0} x^(2^n-1)/(n+1)^s = 1 + x/2^s + x^3/3^s +x^7/4^s+.. may be expressed by the continued fraction: Z(x,s) = [1; f(1)^s/x, -f(2)^s/x, -f(3)^s/x,...,f(n)^s*(-1)^e(n)/x,...] such that the (2^n-1)-th convergent = Sum_{k=0..n} x^(2^k-1)/(k+1)^s, where f(n) = (b(n)+2)/(b(n)+1)^2 and e(n) = A073089(n+1) for n>=1, and b(n) = A007814(n) the exponent of highest power of 2 dividing n. Thus a(n) = (A007814(n) + 2)*(-1)^A073089(n+1) are numerators and A123726(n) = (A007814(n) + 1)^2 are denominators of partial quotients. Case s=1. Sum_{n>=0} x^(2^n-1)/(n+1) = [1; 2/x, -(3/4)/x, -2/x, -(4/9)/x, 2/x, (3/4)/x, -2/x, -(5/16)/x, 2/x, -(3/4)/x, -2/x, (4/9)/x, 2/x, (3/4)/x, -2/x, -(6/25)/x, 2/x, -(3/4)/x, -2/x, -(4/9)/x, 2/x, (3/4)/x, -2/x,...]. Note that (2^n-1)-th convergents exactly equal n-th partial sums: [1;2/x] = 1 + x/2; [1;2/x,-(3/4)/x,-2/x] = 1 + x/2 + x^3/3; [1;2/x,-(3/4)/x,-2/x,-(4/9)/x,2/x,(3/4)/x,-2/x] = 1 +x/2 +x^3/3 +x^7/4. Case s=2. Sum_{n>=0} x^(2^n-1)/(n+1)^2 = [1; 4/x, -(9/16)/x, -4/x, -(16/81)/x, 4/x, (9/16)/x, -4/x, -(25/256)/x, 4/x, -(9/16)/x, -4/x, (16/81)/x, 4/x, (9/16)/x, -4/x, -(36/625)/x, 4/x, -(9/16)/x, -4/x, (16/81)/x, 4/x ...]. Note that (2^n-1)-th convergents exactly equal n-th partial sums: [1;4/x] = 1 + x/4; [1;4/x,-(9/16)/x,-4/x] = 1 + x/4 + x^3/9; [1;4/x,-(9/16)/x,-4/x,-(16/81)/x,4/x,(9/16)/x,-4/x]=1+x/4+x^3/9+x^7/16. Case s=3. Sum_{n>=0} x^(2^n-1)/(n+1)^3 = [1; 8/x,-(27/64)/x,-8/x,-(64/729)/x,8/x, (27/64)/x,-8/x,-(125/4096)/x, 8/x,-(27/64)/x,-8/x, (64/729)/x, 8/x, (27/64)/x,-8/x,-(216/15625)/x, 8/x, -(27/64)/x, -8/x, (64/729)/x ...]. Likewise, the (2^n-1)-th convergents exactly equal n-th partial sums. It is conjectured that these patterns continue for all s. PROG (PARI) {a(n)=numerator(subst(contfrac(sum(m=0, #binary(n), 1/x^(2^m-1)/(m+1)), n+3)[n+1], x, 1))} (PARI) A007814(n) = valuation(n, 2); A073089(n) = { if(n<=1, return(0)); n-=1; my(v=2^valuation(n, 2)); return((0==bitand(n, v<<1)) != (v%2)); }; \\ From A073089 A123725(n) = if(!n, 1, (A007814(n) + 2) * (-1)^A073089(n+1)); \\ Antti Karttunen, Nov 01 2018 CROSSREFS Cf. A123726 (denominators); A007814, A073089, A089080 (unsigned). Sequence in context: A199968 A066482 A324507 * A089080 A085058 A183152 Adjacent sequences: A123722 A123723 A123724 * A123726 A123727 A123728 KEYWORD cofr,frac,sign AUTHOR Paul D. Hanna, Oct 12 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 24 04:28 EDT 2023. Contains 361454 sequences. (Running on oeis4.)