A156020 -id:A156020 - cp's OEIS frontend

A156618 Denominators of Egyptian fraction for Pi-3 whose partial sums are the convergents.

7, -742, 11978, -3740526, 1099482930, -2202719155, 6600663644, -26413901692, 96840976853, -496325469560, 2346251883960, -44006595799206, 1345586183756654, -4127747481719463, 10251870941174304

Offset: 0

Jaume Oliver Lafont, Feb 11 2009

sign

Numerators are all 1.

			3+1/a(0)=22/7
3+1/a(0)+1/a(1)=333/106
3+1/a(0)+1/a(1)+1/a(2)=355/113

Cf. A001466, A014013, A156019, A156020, A002485, A002486.

c0=3; for (k=2,30,m=contfracpnqn(contfrac(Pi,k));c1=m[1,1]/m[2,1];print1(1/(c1-c0),", ");c0=c1;)

A156019 Numerators in an infinite sum for Pi.

Original entry on oeis.org

3, 15, 73, 1, 2, 3, 7, 1, 2, 2, 1, 2, 1, 3, 1, 2, 6, 1, 1, 3, 1, 6, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 2, 6, 1, 2, 3, 1, 1, 1, 45, 22, 2, 1, 1, 24, 2, 1, 2, 1, 2, 4, 2, 8, 5, 1, 1, 1, 2, 7, 1, 3, 1, 7, 4, 7, 3, 3, 9, 9, 1, 18, 3, 15, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1

Offset: 1

Gary W. Adamson and Alexander R. Povolotsky, Feb 01 2009

For k >= 0, define Q(k) = A002485(2k)/A002486(2k) (convergents to Pi that are less than Pi), so Pi = Sum_{k>=1} (Q(k) - Q(k-1)). Then a(n) is the numerator of Q(n) - Q(n-1).

			a(2) = 15 since A002485(4)/A002486(4) = 333/106, A002485(2)/A002486(2) = 3/1, and 333/106 - 3/1 = 15/106 (see table below).
Pi = 3/1 + 15/106 + 73/877203 + 1/2195225334 + 2/17599271777 + 3/360950005720 + 7/17348726394920 + ....
.
  n  Q(n) = A002485(2n)/A002486(2n)  Q(n) - Q(n-1)  a(n)
  -  ------------------------------  -------------  ----
  0       0/1     = 0                     -           -
  1       3/1     = 3                    3/1          3
  2     333/106   = 3.1415094339...     15/106       15
  3  103993/33102 = 3.1415926530...     73/877203    73

Cf. A000796, A002485, A002486, A156020 (denominators).

a(n) = numerator(A002485(2n)/A002486(2n) - A002485(2n-2)/A002486(2n-2)).

More terms from Alexander R. Povolotsky, Sep 01 2009
Edited by Jon E. Schoenfield, Jan 04 2022
More terms from Jinyuan Wang, Jun 29 2022

A157497 Triangle read by rows, A156348 * A127648.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 4, 0, 4, 1, 0, 0, 0, 5, 1, 6, 9, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 8, 0, 16, 0, 0, 0, 8, 1, 0, 18, 0, 0, 0, 0, 0, 9, 1, 10, 0, 0, 25, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 12, 30, 40, 0, 36, 0, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson & Mats Granvik, Mar 01 2009

Row sums = A157020: (1, 3, 4, 9, 6, 22, 8,...)

			First few rows of the triangle =
1;
1, 2;
1, 0, 3;
1, 4, 0, 4;
1, 0, 0, 0, 5;
1, 6, 9, 0, 0, 6;
1, 0, 0, 0, 0, 0, 7;
1, 8, 0, 16, 0, 0, 0, 8;
1, 0, 18, 0, 0, 0, 0, 0, 9;
1, 10, 0, 0, 25, 0, 0, 0, 0, 10;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11;
1, 12, 30, 40, 0, 36, 0, 0, 0, 0, 0, 12;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
1, 14, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 14;
...
Row 4 = (1, 4, 0, 4) = termwise products of (1, 2, 0, 1) and (1, 2, 3, 4)
where (1, 2, 0, 1) = row 4 of triangle A156348.

Cf. A156348, A126648, A156020

Triangle read by rows, A156348 * A127648. A127648 = an infinite lower triangular matrix with (1, 2, 3,...) as the main diagonal and the rest zeros.

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A156618 Denominators of Egyptian fraction for Pi-3 whose partial sums are the convergents.

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A156019 Numerators in an infinite sum for Pi.

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A157497 Triangle read by rows, A156348 * A127648.

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