A054135 a(n) = T(n,1), array T as in A054134.

2, 4, 9, 19, 39, 79, 159, 319, 639, 1279, 2559, 5119, 10239, 20479, 40959, 81919, 163839, 327679, 655359, 1310719, 2621439, 5242879, 10485759, 20971519, 41943039, 83886079, 167772159, 335544319, 671088639, 1342177279

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

From Jianing Song, May 25 2025: (Start)

As Ely Golden noted in A153894, a(n) is the total number of symbols required in the fully-expanded von Neumann definition of ordinal n - 1, where the string "{}" is used to represent the empty set and spaces are ignored. First examples:

0 = {};

1 = {0} = {{}};

2 = {0,1} = {{},{{}}};

3 = {0,1,2} = {{},{{}},{{},{{}}}};

4 = {0,1,2,3} = {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}.

(End)

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Identical to A052549 and A153894 except for initial term.

Cf. A267524.

Programs

Python

print([2]+[(5*2**(n-2) - 1) for n in range(2, 50)]) # Karl V. Keller, Jr., Jun 12 2022

Formula

For n > 2, a(n) = 10*A000225(n-3) + 9 = 10*(2^(n-3) - 1) + 9 = 10*2^(n-3) - 1. - Gerald McGarvey, Aug 25 2004

a(1)=1, a(n) = n + Sum_{i=1..n-1} a(i) for n > 1. - Gerald McGarvey, Sep 06 2004

a(n) = 5*2^(n-2) - 1 for n > 1. - Karl V. Keller, Jr., Jun 12 2022