A123180 Even positions of Sylvester's sequence A000058; the denominators of the (greedy) Egyptian fraction expansion of Cahen's constant.
2, 7, 1807, 10650056950807, 12864938683278671740537145998360961546653259485195807
Offset: 0
Links
- Amiram Eldar, Table of n, a(n) for n = 0..6
- Eugène Cahen, Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues, Nouvelles Annales de Mathématiques, Vol. 10 (1891), pp. 508-514.
- Eric Weisstein's World of Mathematics, Cahen's Constant.
- Wikipedia, Cahen's constant.
Programs
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Mathematica
f[n_] := n*(n-1)*(n*(n-1)+1)+1; a[0] = 2; a[n_] := a[n] = f[a[n-1]]; Array[a, 5, 0] (* Amiram Eldar, Mar 19 2024 *) 1+NestList[#(#+1)(#^2+#+1) &, 1, 4] (* Oliver Seipel, Aug 25 2024 *)
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PARI
a(n)=if(n, my(k=a(n-1));k*=k-1; k*(k+1)+1, 2) \\ Charles R Greathouse IV, Dec 12 2013
Formula
a(n) = a(n-1)*(a(n-1)-1)*(a(n-1)*(a(n-1)-1)+1)+1.
a(n) is approximately k^4^n with k = 1.5979102180318731783... (A077125). - Charles R Greathouse IV, Dec 12 2013
Sum_{n>=0} 1/a(n) = A118227. - Amiram Eldar, Mar 19 2024
Extensions
a(4) from Charles R Greathouse IV, Dec 12 2013