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 A123746 Numerators of partial sums of a series for 1/sqrt(2). 3
 1, 1, 7, 9, 107, 151, 835, 1241, 26291, 40427, 207897, 327615, 3296959, 5293843, 26189947, 42685049, 1666461763, 2749521971, 13266871709, 22115585443, 211386315749, 355490397193, 1684973959237, 2855358497999, 53747636888759 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Denominators are given by A046161(n),n>=0. The alternating sum over central binomial coefficients scaled by powers of 4, r(n):=sum(((-1)^k)*binomial(2*k,k)/4^k,k=0..n) has the limit s:=lim(r(n),n->infinity) = 1/sqrt(2). From the expansion of 1/sqrt(1-x) for |x|<1 which extends to x=-1 due to Abel's limit theorem and the convergence of the series s. See the W. Lang link. (2^n)*n!*r(n) = A003148(n). [From Wolfdieter Lang, Oct 06 2008] REFERENCES Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116.  Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 W. Lang, Rationals and more. FORMULA a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k* binomial(2*k,k)/4^k, n>=0. r(n) = Sum_{k=0..n} (-1)^k*(2*k-1)!!/(2*k)!!, n>=0, with the double factorials A001147 and A000165. EXAMPLE a(3)=9 because r(n)=1-1/2+3/8-5/16 = 9/16 = a(3)/A046161(3). MATHEMATICA r[n_] := Sum[(-1/4)^k*Binomial[2*k, k], {k, 0, n}]; Numerator[Table[ r[n], {n, 0, 30}]] (* G. C. Greubel, Mar 28 2018 *) PROG (PARI) {r(n) = sum(k=0, n, (-1/4)^k*binomial(2*k, k))}; for(n=0, 30, print1(numerator(r(n)), ", ")) \\ G. C. Greubel, Mar 28 2018 CROSSREFS Cf. A120088/(2*A120777) partial sums for a series of sqrt(2). Contribution from Johannes W. Meijer, Nov 23 2009: (Start) Equals A003148 divided by A049606. (End) Sequence in context: A317401 A137058 A116237 * A152551 A012252 A262538 Adjacent sequences:  A123743 A123744 A123745 * A123747 A123748 A123749 KEYWORD nonn,frac,easy AUTHOR Wolfdieter Lang, Nov 10 2006 STATUS approved

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Last modified March 25 03:50 EDT 2019. Contains 321450 sequences. (Running on oeis4.)