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#59 by Michel Marcus at Mon Jul 03 05:14:08 EDT 2023
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#58 by Joerg Arndt at Mon Jul 03 02:57:30 EDT 2023
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#57 by Joerg Arndt at Mon Jul 03 02:57:22 EDT 2023
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#56 by Joerg Arndt at Mon Jul 03 02:57:19 EDT 2023
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Andrea Pinos, <a href="/A005371/a005371_3.txt">Approximate values of large terms</a>
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#55 by Joerg Arndt at Mon Jul 03 02:57:06 EDT 2023
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From Andrea Pinos, May 03 2023: (Start)
a(n) = a(n-1)*a(n-2) + a(n-3)*(-1)^((n-1) mod 3).
a(n) = phi^L(n) - (-1)^((n+1) mod 3) / phi^L(n).
for n==0 (mod 6): a(n) = 2*cosh(log(phi)*2*cosh(n*log(phi)));
for n==1 or n==5 (mod 6): a(n) = 2*sinh(log(phi)*2*sinh(n*log(phi)));
for n==2 or n==4 (mod 6): a(n) = 2*sinh(log(phi)*2*cosh(n*log(phi)));
for n==3 (mod 6): a(n) = 2*cosh(log(phi)*2*sinh(n*log(phi))). (End)
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#54 by Michel Marcus at Sat May 13 06:23:12 EDT 2023
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Discussion
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| Tue May 16
| 09:04
| Kevin Ryde: That convergence, when it does converge, is presumably to one of the solutions to Lucas(x) = x (the continuous version of the Lucas function). I don't find a sequence for that.
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| 09:08
| Kevin Ryde: I still think all the sinh and cosh don't match the problem particularly well.
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| 10:32
| Andrea Pinos: Ok, the convergence isn't so important for the sequence. But the expressions with hyperbolic functions are a method more fast to do this calculus , similar but not equal to Binet's formula
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#53 by Michel Marcus at Sat May 13 06:23:06 EDT 2023
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Andrea Pinos, <a href="/A005371/a005371_3.txt">approximateApproximate values of large terms</a>
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#52 by Andrea Pinos at Sat May 13 05:32:38 EDT 2023
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#51 by Andrea Pinos at Sat May 13 05:23:07 EDT 2023
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Andrea Pinos, <a href="/A005371/a005371_23.txt">forecastapproximate values of large terms</a>
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Discussion
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| Sat May 13
| 05:32
| Andrea Pinos: Ok, I've specified all terms. There is also an interesting thing: with an iteration x->L(x) the function converges to fixed point 0.9762305652308763 for some intervals (for instance 0.68077870383681 < x < 1 ). I've sought something about this. Could be interesting?
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#50 by Andrea Pinos at Fri May 12 06:41:01 EDT 2023
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Discussion
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| Fri May 12
| 20:53
| Kevin Ryde: I'm a bit doubtful about the merit of the sinh and cosh forms. They're presented in A000032 and not sure that expanding them out adds much.
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| 20:59
| Kevin Ryde: The word "forecast" is not right. What's presented is initial decimal digits of large terms (yes?). Maybe "approximate values of large terms", and a note in the file that the decimals are truncated at those shown.
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