This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A113208 #12 Jun 08 2025 04:08:53
%S A113208 1,1,2,4,10,44,1426,17592187106356
%N A113208 Half-fixed-point of ascending descending base exponent transform.
%C A113208 a(9) has 429 digits.
%C A113208 The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. There is no nontrivial integer fixed point of the transform.
%F A113208 a(1) = 1. For n > 1: a(n) = (1/2) * Sum_{i=1..n} a(i)^a(n-i+1).
%e A113208 a(2) = 1 because a(1)^a(2) + a(2)^a(1) = 1^1 + 1^1 = 2 and 2/2 = 1.
%e A113208 a(3) = 2 because a(1)^a(3) + a(2)^a(2) + a(3)^a(1) = 1^2 + 1^1 + 2^1 = 4 and 4/2 = 2.
%e A113208 a(4) = 4 because a(1)^a(4) + a(2)^a(3) + a(3)^a(2) + a(4)^a(1) = 1^4 + 1^2 + 2^1 + 4^1 = 8 and 8/2 = 4.
%e A113208 a(5) = 10 because a(1)^a(5) + a(2)^a(4) + a(3)^a(3) + a(4)^a(2) + a(5)^a(1) = 1^10 + 1^4 + 2^2 + 4^1 + 10^1 = 20 and 20/2 = 10.
%e A113208 a(6) = 44 because 1^44 + 1^10 + 2^4 + 4^2 + 10^1 + 44^1 = 88 and 88/2 = 44.
%e A113208 a(7) = (1^1426 + 1^44 + 2^10 + 4^4 + 10^2 + 44^1 + 1426^1)/2 = 1426.
%e A113208 a(8) = (1^17592187106356 + 1^1426 + 2^44 + 4^10 + 10^4 + 44^2 + 1426^1 + 17592187106356^1)/2 = 17592187106356.
%t A113208 nmax = 9; a[1] = 1; Do[a[n] = a[n] /. Solve[a[n] == Sum[a[i]^a[n - i + 1], {i, 1, n}]/2, a[n]][[1]], {n, 2, nmax}]; Table[a[n], {n, 1, nmax}] (* _Vaclav Kotesovec_, Jun 08 2025 *)
%Y A113208 Cf. A113122, A113153, A113154.
%K A113208 easy,nonn
%O A113208 1,3
%A A113208 _Jonathan Vos Post_, Jan 06 2006