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 A120088 Numerators of partial sums of a series for sqrt(2). 3
 3, 11, 23, 179, 365, 1439, 2911, 46147, 93009, 369605, 743409, 5917879, 11887761, 47365319, 95064943, 3032383331, 6082445497, 24264959593, 48649328861, 388310999293, 778263028691, 3106935548009, 6225306416473 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Involving alternating sums over scaled Catalan numbers, A000108(k)/4^k. From the expansion of sqrt(1+x) = 1 + x*sum((C_k)*(-x/4)^k,k=0..infty)/2, valid for |x|<=1, one finds for x=+1: sqrt(2) = 1 + sum(((-1)^k)*C(k)/4^k,k=0..infty)/2. The denominators are given by 2*A120777(n). The rationals r(n):=1 + (Sum(((-1)^k)*C(k)/4^k))/2,k=0..n), with the Catalan numbers C(n)=A000108(n), are A120088(n)/A120777(n), n>=0. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 W. Lang, Rationals r(n). FORMULA a(n) = numerator(r(n)), with the rationals defined above. EXAMPLE Rationals r(n): [3/2, 11/8, 23/16, 179/128, 365/256, 1439/1024, 2911/2048, 46147/32768,...] MATHEMATICA r[n_]:= 1+Sum[(-1/4)^k*CatalanNumber[k]/2, {k, 0, n}]; Numerator[Table[ r[n], {n, 0, 50}]] (* G. C. Greubel, Mar 27 2018 *) PROG (PARI) {r(n) = 1 + sum(k=0, n, (-1/4)^k*binomial(2*k, k)/(2*(k+1)))}; for(n=0, 30, print1(numerator(r(n)), ", ")) \\ G. C. Greubel, Mar 27 2018 (MAGMA) [Numerator(1 + (&+[(-1/4)^k*Binomial(2*k, k)/(2*(k+1)): k in [0..n]])): n in [0..30]]; // G. C. Greubel, Mar 27 2018 CROSSREFS For similar partial sums with positive terms (not alternating) see rationals A119951/A120069. For the partial sums (sum(((-1)^k)*C(k)/4^k)), k=0..n) see A120788(n)/A120777(n). Sequence in context: A096297 A081857 A168163 * A081737 A005475 A293404 Adjacent sequences:  A120085 A120086 A120087 * A120089 A120090 A120091 KEYWORD nonn,easy,frac AUTHOR Wolfdieter Lang, Jul 20 2006 STATUS approved

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Last modified April 19 03:04 EDT 2019. Contains 322237 sequences. (Running on oeis4.)