A113208 Half-fixed-point of ascending descending base exponent transform.

1, 1, 2, 4, 10, 44, 1426, 17592187106356

Offset: 1

Author

Jonathan Vos Post, Jan 06 2006

Keywords

• easy
• nonn

Comments

a(9) has 429 digits.

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.

Examples

			a(2) = 1 because a(1)^a(2) + a(2)^a(1) = 1^1 + 1^1 = 2 and 2/2 = 1.
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.
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.
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.
a(6) = 44 because 1^44 + 1^10 + 2^4 + 4^2 + 10^1 + 44^1 = 88 and 88/2 = 44.
a(7) = (1^1426 + 1^44 + 2^10 + 4^4 + 10^2 + 44^1 + 1426^1)/2 = 1426.
a(8) = (1^17592187106356 + 1^1426 + 2^44 + 4^10 + 10^4 + 44^2 + 1426^1 + 17592187106356^1)/2 = 17592187106356.
		

Crossrefs

Cf. A113122, A113153, A113154.

Programs

Mathematica

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 *)

Formula

a(1) = 1. For n > 1: a(n) = (1/2) * Sum_{i=1..n} a(i)^a(n-i+1).