A343007 - OEIS

%I #32 May 06 2021 22:56:46

%S 6,13,26,41,62,85,114,145,182,221,266,313,366,421,482,545,614,685,762,

%T 841,926,1013,1106,1201,1302,1405,1514,1625,1742,1861,1986,2113,2246,

%U 2381,2522,2665,2814,2965,3122,3281,3446,3613,3786,3961,4142,4325,4514,4705

%N Relative position of the average value between two consecutive partial sums of the Leibniz formula for Pi.

%C 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F a(1) = 6; a(n) = a(n-1) + r(n), where r(n) = A047550(n) = 4*n - (-1)^n.

%F G.f.: x*(6 + x + x^3)/((1 + x)*(1 - x)^3). - _Jinyuan Wang_, Apr 03 2021

%F From _Stefano Spezia_, Apr 03 2021: (Start)

%F a(n) = (3 + (-1)^(n+1) + 4*n + 4*n^2)/2.

%F a(2*n) = A102083(n).

%F a(2*n-1) = A254527(n). (End)

%e The first several partial sums are as follows:

%e   n      L(n)

%e   -  ------------

%e   1  4.0000000000

%e   2  2.6666666...

%e   3  3.4666666...

%e   4  2.8952380...

%e   5  3.3396825...

%e   6  2.9760461...

%e   7  3.2837384...

%e   8  3.0170718...

%e .

%e 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 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 Rest@ CoefficientList[Series[x (6 + x + x^3)/((1 + x) (1 - x)^3), {x, 0, 48}], x] (* _Michael De Vlieger_, Apr 05 2021 *)

%Y Cf. A047550, A102083, A254527.

%K nonn,easy

%O 1,1

%A _Raphael Ranna_, Apr 02 2021