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A296786 a(1) = a(2) = a(5) = 2, a(3) = 1, a(4) = 3, a(6) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 6.

%I A296786 #20 Feb 20 2019 21:09:43
%S A296786 2,2,1,3,2,5,7,8,7,4,11,12,11,4,15,16,15,4,19,20,19,4,23,24,23,4,27,
%T A296786 28,27,4,31,32,31,4,35,36,35,4,39,40,39,4,43,44,43,4,47,48,47,4,51,52,
%U A296786 51,4,55,56,55,4,59,60,59,4,63,64,63,4,67,68,67,4,71,72,71,4,75,76,75,4,79,80,79,4
%N A296786 a(1) = a(2) = a(5) = 2, a(3) = 1, a(4) = 3, a(6) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 6.
%C A296786 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)). See comments in A296518.
%H A296786 Colin Barker, <a href="/A296786/b296786.txt">Table of n, a(n) for n = 1..1000</a>
%H A296786 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 A296786 <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 A296786 a(4*k-1) = a(4*k+1) = 4*k-1, a(4*k) = 4*k, a(4*k+2) = 4, for k > 1.
%F A296786 From _Colin Barker_, Dec 28 2017: (Start)
%F A296786 G.f.: x*(2 + 2*x + x^2 + 3*x^3 - 2*x^4 + x^5 + 5*x^6 + 2*x^7 + 5*x^8 - 4*x^9 - 2*x^10 - x^11 - x^12 + x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
%F A296786 a(n) = 2*a(n-4) - a(n-8) for n>14.
%F A296786 (End)
%p A296786 a:= proc(n) option remember; procname(n-procname(n-1))+procname(n-procname(n-2))+procname(n-procname(n-3)) end proc:
%p A296786 a(1):= 2: a(2):= 2: a(3):= 1: a(4):= 3: a(5):= 2: a(6):= 5:
%p A296786 map(a, [$1..100]); # after _Robert Israel_ at A296440
%t A296786 a[n_] := a[n] = If[n<7, {2, 2, 1, 3, 2, 5}[[n]], a[n - a[n-1]] + a[n - a[n-2]] + a[n - a[n-3]]]; Array[a, 100] (* after _Giovanni Resta_ at A296440 *)
%o A296786 (PARI) q=vector(10^5); q[1]=2;q[2]=2;q[3]=1;q[4]=3;q[5]=2;q[6]=5;for(n=7, #q, q[n] = q[n-q[n-1]]+q[n-q[n-2]]+q[n-q[n-3]]); q
%o A296786 (PARI) Vec(x*(2 + 2*x + x^2 + 3*x^3 - 2*x^4 + x^5 + 5*x^6 + 2*x^7 + 5*x^8 - 4*x^9 - 2*x^10 - x^11 - x^12 + x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2) + O(x^100)) \\ _Colin Barker_, Dec 29 2017
%Y A296786 Cf. A005185, A244477, A268368, A278055, A296413, A296440, A296518, A296690.
%K A296786 nonn,easy
%O A296786 1,1
%A A296786 _Altug Alkan_, Dec 20 2017