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, 2684354559, 5368709119, 10737418239

Offset

1,1

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

Paolo Xausa, Table of n, a(n) for n = 1..1000

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

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

From Elmo R. Oliveira, Jun 13 2026: (Start)

G.f.: x*(2 - 2*x + x^2)/((1 - 2*x)*(1 - x)).

E.g.f.: (5*exp(2*x) - 4*exp(x) - 1 + 2*x)/4.

a(n) = 3*a(n-1) - 2*a(n-2) for n >= 4. (End)

Mathematica

Ceiling[5*2^(Range[35] - 2) - 1] (* Paolo Xausa, Jun 18 2026 *)

Prog

(Python) print([2]+[(5*2**(n-2) - 1) for n in range(2, 50)]) # Karl V. Keller, Jr., Jun 12 2022
(PARI) a(n)=if(n>1, 5*2^n/4-1, 2) \\ Charles R Greathouse IV, May 30 2026

Crossrefs

Identical to A052549 and A153894 except for initial term.

Cf. A000225, A054134, A267524.

Sequence in context: A265387 A293322 A267157 * A171858 A099568 A018001

Adjacent sequences: A054132 A054133 A054134 * A054136 A054137 A054138

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms from Elmo R. Oliveira, Jun 13 2026

Status

approved