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 A343007 Relative position of the average value between two consecutive partial sums of the Leibniz formula for Pi. 0
 6, 13, 26, 41, 62, 85, 114, 145, 182, 221, 266, 313, 366, 421, 482, 545, 614, 685, 762, 841, 926, 1013, 1106, 1201, 1302, 1405, 1514, 1625, 1742, 1861, 1986, 2113, 2246, 2381, 2522, 2665, 2814, 2965, 3122, 3281, 3446, 3613, 3786, 3961, 4142, 4325, 4514, 4705 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Define L(n) to be the n-th partial sum of the Leibniz formula Pi = 4 - 4/3 + 4/5 - 4/7 + ..., i.e., L(n) = Sum_{j=1..n} 4*(-1)^(j+1)/(2*j-1). For every positive integer n, L(n+1) is closer to Pi than L(n) is. If we let V be the average of the two consecutive partial sums L(n) and L(n+1), then the partial sums that lie closest to V are L(a(n)-1) and L(a(n)+1) (one of which is above V, the other below). LINKS Table of n, a(n) for n=1..48. Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA a(1) = 6; a(n) = a(n-1) + r(n), where r(n) = A047550(n) = 4*n - (-1)^n. G.f.: x*(6 + x + x^3)/((1 + x)*(1 - x)^3). - Jinyuan Wang, Apr 03 2021 From Stefano Spezia, Apr 03 2021: (Start) a(n) = (3 + (-1)^(n+1) + 4*n + 4*n^2)/2. a(2*n) = A102083(n). a(2*n-1) = A254527(n). (End) EXAMPLE The first several partial sums are as follows: n L(n) - ------------ 1 4.0000000000 2 2.6666666... 3 3.4666666... 4 2.8952380... 5 3.3396825... 6 2.9760461... 7 3.2837384... 8 3.0170718... . For n=1, the average of the partial sums L(1) and L(2) is V = (L(1) + L(2))/2 = (4 + 2.6666666...)/2 = 3.3333333...; the two partial sums closest to V are L(5)=3.3396825... and L(7)=3.2837384..., and V lies in the interval between them, so a(1)=6. The formula as it is written works for all data in the sequence, but it needs to be proven that it works for all possible integer values of n. MATHEMATICA Rest@ CoefficientList[Series[x (6 + x + x^3)/((1 + x) (1 - x)^3), {x, 0, 48}], x] (* Michael De Vlieger, Apr 05 2021 *) CROSSREFS Cf. A047550, A102083, A254527. Sequence in context: A162432 A117072 A081395 * A192762 A268721 A183337 Adjacent sequences: A343004 A343005 A343006 * A343008 A343009 A343010 KEYWORD nonn,easy AUTHOR Raphael Ranna, Apr 02 2021 STATUS approved

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Last modified September 18 07:20 EDT 2024. Contains 375996 sequences. (Running on oeis4.)