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A110257 Numerators in the coefficients that form the odd-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x. 7

%I #18 Jun 13 2019 03:30:31

%S 1,5,81,325,20825,83349,1334025,5337189,1366504425,5466528925,

%T 87470372561,349899121845,22394407746529,89580335298125,

%U 1433319858545625,5733391194015525,5871086572691471625

%N Numerators in the coefficients that form the odd-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.

%C Limit a(n)/A110258(n) = limit A110255(2*n-1)/A110256(2*n-1) = 4/Pi.

%H Paul D. Hanna, <a href="/A110257/b110257.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = A110255(2*n-1).

%F a(n) = (4*n+1)*A002894(n)/4^A000120(n). - _Peter Luschny_, Mar 23 2014

%e arctan(1/x) = 1/x - 1/(3*x^3) + 1/(5*x^5) - 1/(7*x^7) +-...

%e = [0; x, 3*x, 5/4*x, 28/9*x, 81/64*x, 704/225*x, 325/256*x, 768/245*x, 20825/16384*x, 311296/99225*x, 83349/65536*x, 1507328/480249*x, 1334025/1048576*x, 3145728/1002001*x,...]

%e = 1/(x + 1/(3*x + 1/(5/4*x + 1/(28/9*x + 1/(81/64*x +...))))).

%e The coefficients of x in the even-indexed partial quotients converge to Pi: {3, 28/9, 704/225, 768/245, 311296/99225, ...}.

%e The coefficients of x in the odd-indexed partial quotients converge to 4/Pi: {1, 5/4, 81/64, 325/256, 20825/16384, ...}.

%p a := n -> (4*n+1)*binomial(2*n,n)^2/4^(add(i,i=convert(n,base,2)));

%p seq(a(n), n=0..16); # _Peter Luschny_, Mar 23 2014

%t a[n_] := (4n+1) Binomial[2n, n]^2 / 4^DigitCount[n, 2, 1];

%t Array[a, 16] (* _Jean-François Alcover_, Jun 13 2019, from Maple *)

%o (PARI) {a(n)=numerator(subst((contfrac( sum(k=0,2*n+1,(-1)^k/x^(2*k+1)/(2*k+1)),2*n+2))[2*n],x,1))}

%Y Cf. A110258 (denominators), A110255/A110256 (continued fraction), A110259/A110260.

%K frac,nonn

%O 1,2

%A _Paul D. Hanna_, Jul 18 2005

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Last modified March 28 16:00 EDT 2024. Contains 371254 sequences. (Running on oeis4.)