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

%I A054135 #23 May 25 2025 20:47:13
%S A054135 2,4,9,19,39,79,159,319,639,1279,2559,5119,10239,20479,40959,81919,
%T A054135 163839,327679,655359,1310719,2621439,5242879,10485759,20971519,
%U A054135 41943039,83886079,167772159,335544319,671088639,1342177279
%N A054135 a(n) = T(n,1), array T as in A054134.
%C A054135 From _Jianing Song_, May 25 2025: (Start)
%C A054135 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:
%C A054135   0 = {};
%C A054135   1 = {0} = {{}};
%C A054135   2 = {0,1} = {{},{{}}};
%C A054135   3 = {0,1,2} = {{},{{}},{{},{{}}}};
%C A054135   4 = {0,1,2,3} = {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}.
%C A054135 (End)
%H A054135 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F A054135 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
%F A054135 a(1)=1, a(n) = n + Sum_{i=1..n-1} a(i) for n > 1. - _Gerald McGarvey_, Sep 06 2004
%F A054135 a(n) = 5*2^(n-2) - 1 for n > 1. - _Karl V. Keller, Jr._, Jun 12 2022
%o A054135 (Python) print([2]+[(5*2**(n-2) - 1) for n in range(2, 50)]) # _Karl V. Keller, Jr._, Jun 12 2022
%Y A054135 Identical to A052549 and A153894 except for initial term.
%Y A054135 Cf. A267524.
%K A054135 nonn
%O A054135 1,1
%A A054135 _Clark Kimberling_