cp's OEIS Frontend

A264758 An eventually quasi-cubic solution to Hofstadter's Q recurrence.

%I A264758 #21 Aug 14 2021 18:43:18
%S A264758 7,0,5,9,3,8,9,2,9,9,3,14,9,3,22,9,2,18,9,3,32,9,3,54,9,2,27,9,3,59,9,
%T A264758 3,113,9,2,36,9,3,95,9,3,208,9,2,45,9,3,140,9,3,348,9,2,54,9,3,194,9,
%U A264758 3,542,9,2,63,9,3,257,9,3,799,9,2,72,9,3,329,9
%N A264758 An eventually quasi-cubic solution to Hofstadter's Q recurrence.
%C A264758 a(n) is the solution to the recurrence relation a(n) = a(n-a(n-1)) + a(n-a(n-2)) [Hofstadter's Q recurrence], with the initial conditions: a(n) = 0 if n <= 0; a(1) = 7, a(2) = 0, a(3) = 5, a(4) = 9, a(5) = 3, a(6) = 8, a(7) = 9, a(8) = 2, a(9) = 9, a(10) = 9, a(11) = 3.
%H A264758 Colin Barker, <a href="/A264758/b264758.txt">Table of n, a(n) for n = 1..1000</a>
%H A264758 Nathan Fox, <a href="http://arxiv.org/abs/1511.06484">Quasipolynomial Solutions to the Hofstadter Q-Recurrence</a>, arXiv preprint arXiv:1511.06484 [math.NT], 2015.
%H A264758 <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,-1).
%F A264758 a(1) = 7, a(2) = 0; thereafter a(9*n) = 9*n, a(9*n+1) = 9, a(9*n+2) = 3, a(9*n+3) = 9/2*n^2+9/2*n+5, a(9*n+4) = 9, a(9*n+5) = 3, a(9*n+6) = 3/2*n^3+9/2*n^2+8*n+8, a(9*n+7) = 9, a(9*n+8) = 2.
%F A264758 From _Colin Barker_, Nov 14 2016: (Start)
%F A264758 G.f.: x*(7 +5*x^2 +9*x^3 +3*x^4 +8*x^5 +9*x^6 +2*x^7 +9*x^8 -19*x^9 +3*x^10 -6*x^11 -27*x^12 -9*x^13 -10*x^14 -27*x^15 -6*x^16 -18*x^17 +15*x^18 -9*x^19 +6*x^20 +27*x^21 +9*x^22 +14*x^23 +27*x^24 +6*x^25 +9*x^26 -x^27 +9*x^28 -5*x^29 -9*x^30 -3*x^31 -3*x^32 -9*x^33 -2*x^34 -2*x^36 -3*x^37) / ((1 -x)^4*(1 +x +x^2)^4*(1 +x^3 +x^6)^4).
%F A264758 a(n) = 4*a(n-9) - 6*a(n-18) + 4*a(n-27) - a(n-36) for n>38.
%F A264758 (End)
%t A264758 CoefficientList[Series[x (7+5x^2+9x^3+3x^4+8x^5+9x^6+2x^7+9x^8- 19x^9+ 3x^10- 6x^11- 27x^12-9x^13- 10x^14- 27x^15-6x^16-18x^17+ 15x^18- 9x^19+6x^20+ 27x^21+9x^22+14x^23+ 27x^24+ 6x^25+ 9x^26-x^27+9x^28- 5x^29-9x^30-3x^31- 3x^32-9x^33-2x^34- 2x^36-3x^37)/((1-x)^4(1+x+x^2)^4 (1+x^3+x^6)^4),{x,0,100}],x] (* _Harvey P. Dale_, Aug 14 2021 *)
%o A264758 (PARI) a = [7,0,5,9,3,8,9,2]; for(n=1, 10, a=concat(a, [9*n, 9, 3, 9/2*n^2+9/2*n+5, 9, 3, 3/2*n^3+9/2*n^2+8*n+8, 9, 2])); a
%Y A264758 Cf. A005185, A188670, A244477, A264756, A264757.
%K A264758 nonn,easy
%O A264758 1,1
%A A264758 _Nathan Fox_, Nov 23 2015