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 A280430 #19 Mar 06 2022 10:47:47
%S A280430 2,4,7,14,19,34,63,76,137,248,282,502,891,980,1718,3000,3233,5594,
%T A280430 9646,10256,17565,30000,31597,53690,91033,95214,160803,271108,282054,
%U A280430 474060,795677,824334,1380142,2308114,2383139,3977364,6631898,6828316,11366227
%N A280430 Longest word T from 2 equal length strings S using no breakpoint reuse.
%C A280430 We start with two strings, of length n, that are the identity permutation of alphabet letters. The space between two adjacent letters is called a breakpoint. We then apply duplications, which take a substring from one string and insert it into the center of the other string. Each duplication uses three breakpoints; two for the substring and one for its destination. Each breakpoint can only be used once. This alternating duplications pattern produces a word T. The formula for the longest possible T uses the length of each string (n).
%D A280430 N. I. Anderson, M. J. Paukner, M. R. Riehl, and A. M. Steinman, String Duplication Histories with No-Breakpoint-Reuse, preprint.
%H A280430 Brona Brejová, Gad M. Landau, Tomáš Vinar, <a href="https://dx.doi.org/10.1098/rsta.2013.0133">Fast computation of a string duplication history under no-breakpoint-reuse</a>, Philos T Roy Soc A 372:20130133.
%H A280430 Jean-Philippe Doyon, Vincent Ranwez, Vincent Daubin, Vincent Berry, <a href="https://doi.org/10.1093/bib/bbr045">Models, algorithms and programs for phylogeny reconciliation</a>, Brief Bioinform, 12:5, 392-400, 2011.
%F A280430 a(n) = n + Sum_{0..k} (((n-2)/(1+Beta^2)) + (2/(Beta-alpha)))alpha^k + (((Beta^2(n-2))/(1+Beta^2)) - (2/(Beta-alpha)))Beta^k + 2 where alpha = (1+sqrt(5))/2, beta = (1-sqrt(5))/2, n = number of letters in each chromosome, and k = floor(2(n-1)/3).
%F A280430 Conjecture: (2 +2*x +3*x^2 -7*x^3 -9*x^4 -6*x^5 +14*x^6 +12*x^7 +7*x^8 -7*x^9 -6*x^10 -3*x^11 +x^13+x^14)/ (1+x+x^2)/ (x-1)^2/ (x^6-3*x^3+1)^2 . - _R. J. Mathar_, Mar 06 2022
%e A280430 In this case, S is 2 strings with length 5. There are 8 breakpoints so 2 duplications can be made. The longest possible T has length 19 which can be obtained using the process below.
%e A280430 ABCDE FGHIJ
%e A280430 A|BCD|E FG|HIJ becomes ABCDE FGBCDHIJ after inserting BCD between G and H.
%e A280430 AB|CDE F|GBCDHI|J becomes ABGBCDHICDE FGBCDHIJ after inserting GBCDHI between B and C.
%t A280430 Table[n + Simplify@ Sum[(((n - 2)/(1 + #2^2)) + (2/(#2 - #1))) #1^k + (((#2^2 (n - 2))/(1 + #2^2)) - (2/(#2 - #1))) #2^k + 2, {k, 0, Floor[2 (n - 1)/3]}] & @@ ((1 + Sqrt[5] {1, -1})/2), {n, 53}] (* _Michael De Vlieger_, Mar 01 2017 *)
%K A280430 nonn
%O A280430 0,1
%A A280430 _Nicole Anderson_, Jan 02 2017
%E A280430 More terms from _Michael De Vlieger_, Mar 01 2017