This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A355288 #30 Mar 22 2025 20:42:45
%S A355288 1,3,7,11,19,31,51,83,135,219,355,575,931,1507,2439,3947,6387,10335,
%T A355288 16723,27059,43783,70843,114627,185471,300099,485571,785671,1271243,
%U A355288 2056915,3328159,5385075,8713235,14098311,22811547,36909859,59721407,96631267,156352675,252983943,409336619,662320563
%N A355288 a(0)=1, a(1)=3, a(2)=7; thereafter a(n) = a(n-1) + a(n-2) + 1.
%C A355288 a(n) is the minimum number of nodes required for a full binary tree of height n with every node height-balanced, and the root node has a balance factor of 0.
%C A355288 Full binary tree: A binary tree is called a full binary tree if each node has exactly two or no children.
%C A355288 Essentially the same as A022403. - _R. J. Mathar_, Sep 23 2022
%H A355288 Sumukh Patel, <a href="/A355288/b355288.txt">Table of n, a(n) for n = 0..1000</a>
%H A355288 Lecture Notes for Computer Science 2530, <a href="https://web.archive.org/web/20240419184618/https://cs.ecu.edu/karl/2530/spr18/Notes/lec37+38.html">Height-balanced trees</a>
%H A355288 NIST, <a href="https://xlinux.nist.gov/dads/HTML/root.html">Root node</a>
%H A355288 Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_tree#Types_of_binary_trees">Full binary tree</a>
%H A355288 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F A355288 a(0)=1, a(1)=3, a(2)=7; thereafter a(n) = a(n-1) + a(n-2) + 1.
%F A355288 From _Stefano Spezia_, Jun 27 2022: (Start)
%F A355288 G.f.: (1 + x + x^2 - 2*x^3)/((1 - x)*(1 - x - x^2)).
%F A355288 a(n) = 2*a(n-1) - a(n-3) for n > 3.
%F A355288 a(n) = 2^(1-n)*((1 + sqrt(5))^(n+1) - (1 - sqrt(5))^(n+1))/sqrt(5) - 1 for n > 0.
%F A355288 E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x) - 2. (End)
%F A355288 a(n) = 4*A000045(n+1) - 1, for n >= 1.
%F A355288 a(n) = 2*A001595(n) + 1, for n >= 1.
%e A355288 The diagrams below illustrate the terms a(3)=11 and a(4)=19.
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%t A355288 Join[{1},Table[4*Fibonacci[n + 1] - 1, {n, 1, 40}]]
%o A355288 (Magma) [n eq 0 select 1 else 4*Fibonacci(n+1) - 1: n in [0..40]];
%Y A355288 Cf. A354902.
%K A355288 nonn,easy
%O A355288 0,2
%A A355288 _Sumukh Patel_, Jun 27 2022