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 A296518 #63 Jan 27 2025 21:11:50
%S A296518 3,1,2,2,1,2,4,8,8,4,8,12,12,4,12,16,16,4,16,20,20,4,20,24,24,4,24,28,
%T A296518 28,4,28,32,32,4,32,36,36,4,36,40,40,4,40,44,44,4,44,48,48,4,48,52,52,
%U A296518 4,52,56,56,4,56,60,60,4,60,64,64,4,64,68,68,4,68,72,72,4,72,76,76,4,76,80,80,4,80,84,84,4,84,88,88,4
%N A296518 a(1) = 3, a(2) = a(5) = 1, a(3) = a(4) = a(6) = 2; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 6.
%C A296518 A quasi-periodic solution to the three-term Hofstadter recurrence a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)). For a quasi-quadratic solution, see also A268368 that uses the convention that evaluating at a nonpositive index gives zero (this sequence and A244477 do not use this convention). See page 119 at the Fox reference for the detailed analysis of solutions with initial conditions with 1 through N. The three-term Hofstadter recurrence also has a slow solution (A278055) and chaotic solutions such as A292351, A296413 and A296440. So the recurrence a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) has all known types of behaviors that are mentioned on page 7 of the Tanny reference in the Links section.
%H A296518 Colin Barker, <a href="/A296518/b296518.txt">Table of n, a(n) for n = 1..1000</a>
%H A296518 Altug Alkan, <a href="https://doi.org/10.1515/math-2018-0124">On a conjecture about generalized Q-recurrence</a>, Open Mathematics (2018) Vol. 16, Issue 1, 1490-1500.
%H A296518 Nathan Fox, <a href="http://sites.math.rutgers.edu/~nhf12/Thesis.pdf">An exploration of nested recurrences using experimental mathematics</a>, Ph.D. thesis, 2017 (19 MB).
%H A296518 Steve Tanny, <a href="/A296518/a296518.pdf">An Invitation to Nested Recurrence Relations</a>, Slides of talk presented at CanaDAM 2013.
%H A296518 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,2,0,0,0,-1).
%F A296518 a(4*k) = a(4*k+1) = a(4*k+3) = 4*k, a(4*k+2) = 4 for k > 1.
%F A296518 From _Colin Barker_, Dec 14 2017: (Start)
%F A296518 G.f.: x*(3 + x + 2*x^2 + 2*x^3 - 5*x^4 + 4*x^7 + 9*x^8 + x^9 + 2*x^10 - 2*x^11 - 3*x^12 - 2*x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
%F A296518 a(n) = 2*a(n-4) - a(n-8) for n > 14.
%F A296518 (End)
%p A296518 a:= proc(n) option remember; procname(n-procname(n-1))+procname(n-procname(n-2))+procname(n-procname(n-3)) end proc:
%p A296518 a(1):= 3: a(2):= 1: a(3):= 2: a(4):= 2: a(5):= 1: a(6):= 2:
%p A296518 map(a, [$1..100]); # after _Robert Israel_ at A296440
%t A296518 Fold[Append[#1, #1[[#2 - #1[[#2 - 1]] ]] + #1[[#2 - #1[[#2 - 2]] ]] + #1[[#2 - #1[[#2 - 3]] ]] ] &, {3, 1, 2, 2, 1, 2}, Range[7, 90]] (* or *)
%t A296518 Rest@ CoefficientList[Series[x (3 + x + 2 x^2 + 2 x^3 - 5 x^4 + 4 x^7 + 9 x^8 + x^9 + 2 x^10 - 2 x^11 - 3 x^12 - 2 x^13)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 90}], x] (* _Michael De Vlieger_, Dec 14 2017 *)
%t A296518 LinearRecurrence[{0,0,0,2,0,0,0,-1},{3,1,2,2,1,2,4,8,8,4,8,12,12,4},100] (* _Harvey P. Dale_, May 30 2018 *)
%o A296518 (PARI) my(q=vector(100)); q[1]=3;q[2]=1;q[3]=2;q[4]=2;q[5]=1;q[6]=2;for(n=7, #q, q[n] = q[n-q[n-1]]+q[n-q[n-2]]+q[n-q[n-3]]); q
%o A296518 (PARI) Vec(x*(3 + x + 2*x^2 + 2*x^3 - 5*x^4 + 4*x^7 + 9*x^8 + x^9 + 2*x^10 - 2*x^11 - 3*x^12 - 2*x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2) + O(x^100)) \\ _Colin Barker_, Dec 14 2017
%o A296518 (Scheme)
%o A296518 ;; With memoization-macro definec.
%o A296518 (definec (A296518 n) (cond ((= 1 n) 3) ((or (= 2 n) (= 5 n)) 1) ((<= n 6) 2) (else (+ (A296518 (- n (A296518 (- n 1)))) (A296518 (- n (A296518 (- n 2)))) (A296518 (- n (A296518 (- n 3)))))))) ;; _Antti Karttunen_, Dec 16 2017
%Y A296518 Cf. A005185, A244477, A268368, A278055, A296413, A296440.
%K A296518 nonn,easy
%O A296518 1,1
%A A296518 _Altug Alkan_, Dec 14 2017