Göbel's sequence - Alchetron, The Free Social Encyclopedia

In mathematics, Göbel's sequence is a sequence of rational numbers defined by the recurrence relation

x n = 1 + x 0 2 + x 1 2 + ⋯ + x n − 1 2 n ,

with starting value

x 0 = 1.

Göbel's sequence starts with

1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ... (sequence A003504 in the OEIS)

The first non-integral value is x₄₃.

Generalization

Göbel's sequence can be generalized to kth powers by

x n = 1 + x 0 k + x 1 k + ⋯ + x n − 1 k n .

The least indices at which the k-Göbel sequences assume a non-integral value are

43, 89, 97, 214, 19, 239, 37, 79, 83, 239, ... (sequence A108394 in the OEIS)

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