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A309554 a(1) = a(6) = 1, a(2) = a(3) = a(8) = 2, a(4) = a(7) = 7, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-3)) for n > 8.

%I A309554 #38 Sep 08 2022 08:46:21
%S A309554 1,2,2,7,5,1,7,2,9,3,11,3,6,4,14,5,9,16,6,15,6,10,8,21,21,21,2,28,3,
%T A309554 30,3,6,4,33,5,9,35,6,34,6,10,8,40,40,40,2,47,3,49,3,6,4,52,5,9,54,6,
%U A309554 53,6,10,8,59,59,59,2,66,3,68,3,6,4,71,5,9,73,6,72,6,10,8,78,78,78,2,85,3,87,3,6,4
%N A309554 a(1) = a(6) = 1, a(2) = a(3) = a(8) = 2, a(4) = a(7) = 7, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-3)) for n > 8.
%C A309554 A well-defined solution sequence for recurrence a(n) = a(n-a(n-1)) + a(n-a(n-3)).
%H A309554 Colin Barker, <a href="/A309554/b309554.txt">Table of n, a(n) for n = 1..1000</a>
%H A309554 <a href="/index/Rec#order_38">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).
%F A309554 For k >= 1:
%F A309554 a(19*k-11) = 2,
%F A309554 a(19*k-10) = 19*k-10,
%F A309554 a(19*k-9) = 3,
%F A309554 a(19*k-8) = 19*k-8,
%F A309554 a(19*k-7) = 3,
%F A309554 a(19*k-6) = 6,
%F A309554 a(19*k-5) = 4,
%F A309554 a(19*k-4) = 19*k-5,
%F A309554 a(19*k-3) = 5,
%F A309554 a(19*k-2) = 9,
%F A309554 a(19*k-1) = 19*k-3,
%F A309554 a(19*k) = 6,
%F A309554 a(19*k+1) = 19*k-4,
%F A309554 a(19*k+2) = 6,
%F A309554 a(19*k+3) = 10,
%F A309554 a(19*k+4) = 8,
%F A309554 a(19*k+5) = a(19*k+6) = a(19*k+7) = 19*k+2.
%F A309554 From _Colin Barker_, Aug 08 2019: (Start)
%F A309554 G.f.: x*(1 + x)*(1 + x + x^2 + 6*x^3 - x^4 + 2*x^5 + 5*x^6 - 3*x^7 + 12*x^8 - 9*x^9 + 20*x^10 - 17*x^11 + 23*x^12 - 19*x^13 + 33*x^14 - 28*x^15 + 37*x^16 - 21*x^17 + 27*x^18 - 14*x^19 + 16*x^20 - 10*x^21 + 4*x^22 + 7*x^23 + 12*x^24 - 5*x^25 + 3*x^26 + 7*x^27 - 10*x^28 + 18*x^29 - 21*x^30 + 15*x^31 - 19*x^32 + 24*x^33 - 29*x^34 + 20*x^35 - 17*x^36 + 11*x^37 - 6*x^38 + 2*x^39 - 10*x^40 + 9*x^41 - 6*x^42 + 5*x^43) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18)^2).
%F A309554 a(n) = 2*a(n-19) - a(n-38) for n > 45.
%F A309554 (End)
%o A309554 (PARI) q=vector(100); q[1]=q[6]=1; q[2]=q[3]=q[8]=2; q[4]=q[7]=7; q[5]=5; for(n=9, #q, q[n]=q[n-q[n-1]]+q[n-q[n-3]]); q
%o A309554 (PARI) Vec(x*(1 + x)*(1 + x + x^2 + 6*x^3 - x^4 + 2*x^5 + 5*x^6 - 3*x^7 + 12*x^8 - 9*x^9 + 20*x^10 - 17*x^11 + 23*x^12 - 19*x^13 + 33*x^14 - 28*x^15 + 37*x^16 - 21*x^17 + 27*x^18 - 14*x^19 + 16*x^20 - 10*x^21 + 4*x^22 + 7*x^23 + 12*x^24 - 5*x^25 + 3*x^26 + 7*x^27 - 10*x^28 + 18*x^29 - 21*x^30 + 15*x^31 - 19*x^32 + 24*x^33 - 29*x^34 + 20*x^35 - 17*x^36 + 11*x^37 - 6*x^38 + 2*x^39 - 10*x^40 + 9*x^41 - 6*x^42 + 5*x^43) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18)^2) + O(x^90)) \\ _Colin Barker_, Aug 11 2019
%o A309554 (Magma) I:=[1,2,2,7,5,1,7,2];[n le 8 select I[n] else Self(n-Self(n-1))+Self(n-Self(n-3)): n in [1..90]]; // _Marius A. Burtea_, Aug 08 2019
%Y A309554 Cf. A046700, A244477, A296518, A309492, A309494, A309496.
%K A309554 nonn,easy
%O A309554 1,2
%A A309554 _Altug Alkan_ and _Rémy Sigrist_, Aug 07 2019