A343007 Relative position of the average value between two consecutive partial sums of the Leibniz formula for Pi.

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

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