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%I #12 Sep 08 2022 08:45:25
%S 3,11,23,179,365,1439,2911,46147,93009,369605,743409,5917879,11887761,
%T 47365319,95064943,3032383331,6082445497,24264959593,48649328861,
%U 388310999293,778263028691,3106935548009,6225306416473
%N Numerators of partial sums of a series for sqrt(2).
%C Involving alternating sums over scaled Catalan numbers, A000108(k)/4^k.
%C 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.
%C The denominators are given by 2*A120777(n).
%C 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.
%H G. C. Greubel, <a href="/A120088/b120088.txt">Table of n, a(n) for n = 0..1000</a>
%H W. Lang, <a href="/A120088/a120088.txt">Rationals r(n).</a>
%F a(n) = numerator(r(n)), with the rationals defined above.
%e Rationals r(n): [3/2, 11/8, 23/16, 179/128, 365/256, 1439/1024, 2911/2048, 46147/32768,...]
%t 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 *)
%o (PARI) {r(n) = 1 + sum(k=0,n, (-1/4)^k*binomial(2*k,k)/(2*(k+1)))};
%o for(n=0,30, print1(numerator(r(n)), ", ")) \\ _G. C. Greubel_, Mar 27 2018
%o (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
%Y For similar partial sums with positive terms (not alternating) see rationals A119951/A120069.
%Y For the partial sums (sum(((-1)^k)*C(k)/4^k)), k=0..n) see A120788(n)/A120777(n).
%K nonn,easy,frac
%O 0,1
%A _Wolfdieter Lang_, Jul 20 2006
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