lprinted'...

3*(1/3*(19+3*33^(1/2))^(1/3)+1/3*(19-3*33^(1/2))^(1/3)+1/3)^n/((586+102*33^(1
/2))^(2/3)+4-2*(586+102*33^(1/2))^(1/3))*(586+102*33^(1/2))^(1/3);

This formula has 2 parts, first the numerator is the root of (x^3-x^2-x-1)
no surprise here, but the denominator was obtained using LLL (Pari-Gp)
algorithm. The thing is, if you try to get a closed formula by doing
the Z-transform or anything classical, it won't work very well since
the actual symbolic expression will be huge and won't simplify.

The numerical values of Tribonacci numbers are c**n essentially and
the c here is one of the roots of (x^3-x^2-x-1), then there is another
constant c2. So the exact formula is c**n/c2.

Another way of doing 'exact formulas' are given by using [ ] function
the n'th term of the series expansion of 1/(1+x+x**2) is

1 - 2 floor(1/3 n + 2/3) + floor(1/3 n + 1/3) + floor(1/3 n)

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The twin primes constant.

0.660161815846869573927812110014555778432623

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The Varga constant, also known to be the 1/(one-ninth constant).

9.2890254919208189187554494359517450610317

One-ninth constant is 0.1076539192264845766153234450909471905879765038

-----------------------------------------------------------------------------

0.4749493799879206503325046363279829685595493732172029822833310248
6455792917488386027427564125050214441890378494262395464775250455
2099778523950882780814821592082565202912193041770281959987798787
6404342380353179170625016170252803841553681975679189489592083858

to 256 digits is also this closed expression.

2**(5/4)*sqrt(Pi)*exp(Pi/8)*GAMMA(1/4)**(-2);

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-Zeta(1,1/2).

is also equal to -Zeta(1/2)*(1/2*gamma+1/2*ln(8*Pi)+1/4*Pi).

3.922646139209151727471531446714599513730323971506505209568298485
2547208031503382848806505231041456914038034379886764996843321856
0187370796648866325531877003002927708284792679262934379740474743
4560678349258709176744625306684542186046544092107149397014020908

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-Zeta(-1/2) to 256 digits.

0.2078862249773545660173067253970493022262685312876725376101135571
0614729193229234048754326694073321564310997561412868956566132691
46944583119657056232941095310616400178