This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A343007 #32 May 06 2021 22:56:46
%S A343007 6,13,26,41,62,85,114,145,182,221,266,313,366,421,482,545,614,685,762,
%T A343007 841,926,1013,1106,1201,1302,1405,1514,1625,1742,1861,1986,2113,2246,
%U A343007 2381,2522,2665,2814,2965,3122,3281,3446,3613,3786,3961,4142,4325,4514,4705
%N A343007 Relative position of the average value between two consecutive partial sums of the Leibniz formula for Pi.
%C A343007 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).
%H A343007 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A343007 a(1) = 6; a(n) = a(n-1) + r(n), where r(n) = A047550(n) = 4*n - (-1)^n.
%F A343007 G.f.: x*(6 + x + x^3)/((1 + x)*(1 - x)^3). - _Jinyuan Wang_, Apr 03 2021
%F A343007 From _Stefano Spezia_, Apr 03 2021: (Start)
%F A343007 a(n) = (3 + (-1)^(n+1) + 4*n + 4*n^2)/2.
%F A343007 a(2*n) = A102083(n).
%F A343007 a(2*n-1) = A254527(n). (End)
%e A343007 The first several partial sums are as follows:
%e A343007 n L(n)
%e A343007 - ------------
%e A343007 1 4.0000000000
%e A343007 2 2.6666666...
%e A343007 3 3.4666666...
%e A343007 4 2.8952380...
%e A343007 5 3.3396825...
%e A343007 6 2.9760461...
%e A343007 7 3.2837384...
%e A343007 8 3.0170718...
%e A343007 .
%e A343007 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.
%e A343007 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.
%t A343007 Rest@ CoefficientList[Series[x (6 + x + x^3)/((1 + x) (1 - x)^3), {x, 0, 48}], x] (* _Michael De Vlieger_, Apr 05 2021 *)
%Y A343007 Cf. A047550, A102083, A254527.
%K A343007 nonn,easy
%O A343007 1,1
%A A343007 _Raphael Ranna_, Apr 02 2021