user
The Almkvist-Giullera formula for calculating 

1

π

2
{\displaystyle {\displaystyle {\frac {1}{\pi ^{2}}}}}
 is based on the Calabi-Yau
differential equations of order 4 and 5,   which were originally used to describe certain manifolds
in string theory.

The formula is:

:   1

    π

    2
    =
    32

    ∑

    n
    =
    0

    ∞

    (
    6
    n
    )
    !

    3
    ⋅
    n

    !

    6
    (
    532

    n

    2
    +
    126
    n
    +
    9
    )


    1

    10

    6
    n
    +
    3
    {\displaystyle {\displaystyle {\frac {1}{\pi ^{2}}}=32\sum \_{n=0}^{\infty }{\frac {(6n)!}{3\cdot n!^{6}}}(532n^{2}+126n+9){\frac {1}{10^{6n+3}}}}}

This formula can be used to calculate the constant 

1

π

2
{\displaystyle {\displaystyle {\frac {1}{\pi ^{2}}}}}
,   and thus to calculate 

π
{\displaystyle {\displaystyle \pi }}
.

Note that, because the product of all terms but the power of 1000 can be calculated as an integer,
the terms in the series can be separated into a large integer term:

:   32
    (
    6
    n
    )
    !

    3
    ⋅
    n

    !

    6
    (
    532

    n

    2
    +
    126
    n
    +
    9
    )
    {\displaystyle {\displaystyle {\frac {32(6n)!}{3\cdot n!^{6}}}(532n^{2}+126n+9)}}

multiplied by a negative integer power of 10:

:   10

    −
    (
    6
    n
    +
    3
    )
    {\displaystyle {\displaystyle 10^{-(6n+3)}}}

Task
:   * Print the integer portions (the starred formula, which is without the power of 1000 divisor) of the first 10 terms of the series.
    * Use the complete formula to calculate and print 

      π
      {\displaystyle {\displaystyle \pi }}
       to 70 decimal digits of precision.

Reference
:   * Gert Almkvist and Jesús Guillera, Ramanujan-like series for 1/π2 and string theory, Experimental Mathematics, 21 (2012), page 2, formula 1.
assistant
```
# 20201013 Raku programming solution

use BigRoot;
use Rat::Precise;
use experimental :cached;

BigRoot.precision = 75 ;
my $Precision     = 70 ;
my $AGcache       =  0 ;

sub postfix:<!>(Int $n --> Int) is cached { [*] 1 .. $n }

sub Integral(Int $n --> Int) is cached {
   (2⁵*(6*$n)! * (532*$n² + 126*$n + 9)) div (3*($n!)⁶)
}

sub A-G(Int $n --> FatRat) is cached { # Almkvist-Giullera
   Integral($n).FatRat / (10**(6*$n + 3)).FatRat
}

sub Pi(Int $n --> Str) {
   (1/(BigRoot.newton's-sqrt: $AGcache += A-G $n)).precise($Precision)
}

say "First 10 integer portions : ";
say $_, "\t", Integral $_ for ^10;

my $target = Pi my $Nth = 0;

loop { $target eq ( my $next = Pi ++$Nth ) ?? ( last ) !! $target = $next }

say "π to $Precision decimal places is :\n$target"

```

Output:

```
First 10 integer portions :
0       96
1       5122560
2       190722470400
3       7574824857600000
4       312546150372456000000
5       13207874703225491420651520
6       567273919793089083292259942400
7       24650600248172987140112763715584000
8       1080657854354639453670407474439566400000
9       47701779391594966287470570490839978880000000
π to 70 decimal places is :
3.1415926535897932384626433832795028841971693993751058209749445923078164
```