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%I #13 Sep 08 2022 08:45:27
%S 3,53,433,13941,111759,1789509,14320329,916611309,7333257267,
%T 117334608047,938685468127,30038055403185,240304869286059,
%U 3844880951681069,30759058568289097,3937160132112061181
%N Numerators of partial sums of a series used for the series of sqrt(2) + sqrt(3) involving Catalan numbers.
%C The corresponding denominators are A120785(n).
%C The limit of this series is 4*(4-(sqrt(2)+sqrt(3))) = 3.414942518 (maple10, 10 digits).
%C See A121503 for the geometric interpretation of sqrt(2)+sqrt(3) and a Popper reference.
%H G. C. Greubel, <a href="/A121504/b121504.txt">Table of n, a(n) for n = 0..830</a>
%H W. Lang, <a href="/A121504/a121504.txt">Rationals r(n), limit.</a>
%F a(n) = numerator(r(n)) with r(n):= sum(C(k)*(1+2^(k+1))/16^k,k=0..n), n>=0, with C(k)=A000108(k) (Catalan numbers).
%e Rationals r(n): [3, 53/16, 433/128, 13941/4096, 111759/32768,
%e 1789509/524288, 14320329/4194304, 916611309/268435456,...].
%t Table[Numerator[Sum[CatalanNumber[k]*(1 + 2^(k + 1))/16^k, {k, 0, n}]], {n, 0, 50}] (* _G. C. Greubel_, Sep 27 2018 *)
%o (PARI) for(n=0, 30, print1(numerator(sum(k=0,n, binomial(2*k,k)*(1 + 2^(k+1))/(16^k*(k+1)))), ", ")) \\ _G. C. Greubel_, Sep 27 2018
%o (Magma) [Numerator( (&+[Binomial(2*k,k)*(1 + 2^(k+1))/(16^k*(k+1)): k in [0..n]]) ): n in [0..30]]; // _G. C. Greubel_, Sep 27 2018
%Y A121503/(4*A120785) are the partial sums of a series for sqrt(2)+sqrt(3).
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Aug 16 2006
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