A144836 a(n) = round(phi^(4^n)) where phi is the golden ratio (A001622).
2, 7, 2207, 23725150497407, 316837008400094222150776738483768236006420971486980607
Offset: 0
Links
- Amiram Eldar, Table of n, a(n) for n = 0..6
Programs
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Maple
a := proc(n) option remember; if n = 0 then 2 elif n = 1 then 7 else a(n-1)^4 - 4*a(n-1)^2 + 2 end if; end proc: seq(a(n), n = 0..4); # Peter Bala, Nov 28 2022
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Mathematica
Table[Round[GoldenRatio^(4^n)], {n, 0, 5}] c = (1 + Sqrt[5])/2; Join[{2}, Table[Expand[c^(4^n) + (1 - c)^(4^n)], {n, 1, 5}]] (* Artur Jasinski, Oct 05 2008 *) Table[Round[2*Cosh[4^n*ArcCosh[Sqrt[5]/2]]], {n, 0, 5}] (* Artur Jasinski, Oct 09 2008 *) a[n_] := LucasL[4^n]; a[0] = 2; Array[a, 5, 0] (* Amiram Eldar, Jul 12 2025 *) -
PARI
a(n)=round(((1+sqrt(5))/2)^4^n) \\ Charles R Greathouse IV, Jul 29 2011
Formula
a(n) = Lucas(4^n) = A000032(4^n), n>0.
a(n) = phi^(4^n) + (1 - phi)^(4^n) = phi^(4^n) + (-phi)^(-4^n), where phi is golden ratio = (1 + sqrt(5))/2 = 1.6180339887..., n>0. - Artur Jasinski, Oct 05 2008
a(n) = 2*cosh(4^n*arccosh(sqrt(5)/2)), n>0. - Artur Jasinski, Oct 09 2008
a(n+1) = a(n)^4 - 4*a(n-1)^2 + 2 with a(1) = 7. - Peter Bala, Nov 28 2022
Extensions
Offset corrected by Charles R Greathouse IV, May 15 2013
Offset changed to 0 by Georg Fischer, Sep 02 2022
New name from Peter Bala, Nov 18 2022
Revised by editors, Jul 12 2025