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 A105531 Decimal expansion of arctan 1/3. 6
 3, 2, 1, 7, 5, 0, 5, 5, 4, 3, 9, 6, 6, 4, 2, 1, 9, 3, 4, 0, 1, 4, 0, 4, 6, 1, 4, 3, 5, 8, 6, 6, 1, 3, 1, 9, 0, 2, 0, 7, 5, 5, 2, 9, 5, 5, 5, 7, 6, 5, 6, 1, 9, 1, 4, 3, 2, 8, 0, 3, 0, 5, 9, 3, 5, 6, 7, 5, 6, 2, 3, 7, 4, 0, 5, 8, 1, 0, 5, 4, 4, 3, 5, 6, 4, 0, 8, 4, 2, 2, 3, 5, 0, 6, 4, 1, 3, 7, 4, 4, 3, 9, 0, 0, 7 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS arctan(1/3) + A073000 = 2*arctan(1/3) + A105533 = Pi/4. LINKS P. Bala, New series for old functions Kunle Adegoke, Infinite arctangent sums involving Fibonacci and Lucas numbers, arXiv:1603.08097 [math.NT], 2016. Eric Weisstein's World of Mathematics, Machin-Like Formulas from Mathworld FORMULA From Peter Bala, Feb 04 2015: (Start) arctan(1/3) = 1/3*Sum {k >= 0} (-1)^k/((2*k + 1)*9^k). Define a pair of integer sequences A(n) = 9^n*(2*n + 1)!/n! and B(n) = A(n)*sum {k = 0..n} (-1)^k/((2*k + 1)*9^k). Both sequences satisfy the same recurrence equation u(n) = (32*n + 20)*u(n-1) + 36*(2*n - 1)^2*u(n-2). From this observation we find the continued fraction expansion arctan(1/3) = 1/3*(1 - 2/(54 + 36*3^2/(84 + 36*5^2/(116 + ... + 36*(2*n - 1)^2/((32*n + 20) + ... ))))). arctan(1/3) = 3/10 * Sum {k >= 0} (2/5)^k/( (2*k + 1)*binomial(2*k,k) ). Define a pair of integer sequences C(n) = 10^n*(2*n + 1)!/n! and D(n) = C(n)*sum {k = 0..n} (2/5)^k/( (2*k + 1)*binomial(2*k,k) ). Both sequences satisfy the same recurrence equation u(n) = (44*n + 20)*u(n-1) - 80*n*(2*n - 1)*u(n-2). From this observation we obtain the continued fraction expansion arctan(1/3) = 3/10*( 1 + 4/(60 - 480/(108 - 1200/(152 - ... - 80*n*(2*n - 1)/((44*n + 20) - ... ))))). (End) arctan(1/3) = Sum{k>=0} arctan((L(4k+2)/F(4k+2)^2) where L=A000032 and F=A000045. See also A033890 and A246453. - Michel Marcus, Mar 29 2016 From Amiram Eldar, Aug 09 2020: (Start) Equals Sum_{k>=2} arctan(1/(2*k^2)) = Sum_{k>=2} (-1)^k arctan(2/k^2). Equals Integral{x=1..2} 1/(x^2 + 1) dx. (End) EXAMPLE 0.3217505543966421934014046143... MATHEMATICA RealDigits[ArcTan[1/3], 10, 120][] (* Harvey P. Dale, Oct 28 2011 *) PROG (PARI) atan(1/3) \\ Michel Marcus, Mar 29 2016 CROSSREFS Sequence in context: A016556 A067050 A001355 * A129689 A115990 A277919 Adjacent sequences:  A105528 A105529 A105530 * A105532 A105533 A105534 KEYWORD cons,nonn AUTHOR Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005 STATUS approved

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Last modified January 21 12:45 EST 2021. Contains 340350 sequences. (Running on oeis4.)