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 A294732 #40 Feb 21 2023 10:38:22
%S A294732 1,2,3,5,6,8,9,11,12,14,15,17,18,20,21,23,24,26,27,29,30,32,33,35,36,
%T A294732 38,39,41,42,44,45,47,48,50,51,53,54,56,57,59,60,62,63,65,66,68,69,71,
%U A294732 72,74,75,77,78,80,81,83,84,86,87,89,90,92,93,95,96,98,99
%N A294732 Maximal diameter of the connected cubic graphs on 2*n vertices.
%H A294732 Colin Barker, <a href="/A294732/b294732.txt">Table of n, a(n) for n = 2..1000</a>
%H A294732 L. Caccetta and W. F. Smyth, <a href="https://doi.org/10.1016/0012-365X(92)90047-J">Graphs of maximum diameter</a>, Discrete Mathematics, Volume 102, Issue 2, 20 May 1992, Pages 121-141.
%H A294732 Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cubics/index.html">Cubic Graphs</a>, October 1996. [Broken link]
%H A294732 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A294732 From _Andrew Howroyd_, Dec 15 2017: (Start)
%F A294732 a(n) = (6*n - 11 - (-1)^n)/4 for n > 2.
%F A294732 a(n) = a(n-2) + 3 for n > 4. (End)
%F A294732 From _Colin Barker_, Dec 16 2017: (Start)
%F A294732 G.f.: x^2*(1 + x + x^3) / ((1 - x)^2*(1 + x)).
%F A294732 a(n) = a(n-1) + a(n-2) - a(n-3) for n > 5. (End)
%F A294732 a(n) = A267528(n-1). - _Hugo Pfoertner_, Oct 10 2018
%F A294732 E.g.f.: (6 + 2*x + x^2 + 3*(x - 2)*cosh(x) + (3*x - 5)*sinh(x))/2. - _Stefano Spezia_, Feb 20 2023
%e A294732 a(5)=5 because there exists a unique graph (up to permutations) on 2*5=10 nodes, given by the following adjacency matrix
%e A294732 .
%e A294732 1 2 3 4 5 6 7 8 9 10
%e A294732 1 . 1 1 1 . . . . . .
%e A294732 2 1 . 1 1 . . . . . .
%e A294732 3 1 1 . . 1 . . . . .
%e A294732 4 1 1 . . 1 . . . . .
%e A294732 5 . . 1 1 . 1 . . . .
%e A294732 6 . . . . 1 . 1 1 . .
%e A294732 7 . . . . . 1 . . 1 1
%e A294732 8 . . . . . 1 . . 1 1
%e A294732 9 . . . . . . 1 1 . 1
%e A294732 10 . . . . . . 1 1 1 .
%e A294732 .
%e A294732 that requires a shortest possible walk using 5 edges to get from node 1 to node 9.
%e A294732 From _Andrew Howroyd_, Dec 15 2017: (Start)
%e A294732 The following constructions are optimal (see theorem 5 of Caccetta et al.).
%e A294732 Pattern for odd n >= 10. Each additional 4 nodes increases diameter by 3.
%e A294732 o o o o
%e A294732 / | \ / | \ / | \ / | \
%e A294732 o--o o---o | o---o | o---o o--o
%e A294732 \ | / \ | / \ | / \ | /
%e A294732 o o o o
%e A294732 Pattern for even n >= 12. Each additional 4 nodes increases diameter by 3.
%e A294732 o--o o o o
%e A294732 / | | \ / | \ / | \ / | \
%e A294732 o--o | o---o | o---o | o---o o--o
%e A294732 \ | | / \ | / \ | / \ | /
%e A294732 o--o o o o
%e A294732 (End)
%t A294732 Prepend[LinearRecurrence[{1, 1, -1}, {2, 3, 5}, 100], 1] (* _Jean-François Alcover_, Dec 27 2017 *)
%o A294732 (PARI) Vec(x^2*(1 + x + x^3) / ((1 - x)^2*(1 + x)) + O(x^100)) \\ _Colin Barker_, Dec 16 2017
%Y A294732 Cf. A002851, A204329, A296525, A114119, A045506, A007494.
%Y A294732 Apart from initial term, duplicate of A267528.
%K A294732 nonn,easy
%O A294732 2,2
%A A294732 _Hugo Pfoertner_, Dec 13 2017
%E A294732 Terms a(12) and beyond from _Andrew Howroyd_, Dec 15 2017