A264757 An eventually quasi-quadratic solution to Hofstadter's Q recurrence.

4, 0, 5, 6, 2, 6, 6, 3, 11, 6, 2, 12, 6, 3, 23, 6, 2, 18, 6, 3, 41, 6, 2, 24, 6, 3, 65, 6, 2, 30, 6, 3, 95, 6, 2, 36, 6, 3, 131, 6, 2, 42, 6, 3, 173, 6, 2, 48, 6, 3, 221, 6, 2, 54, 6, 3, 275, 6, 2, 60, 6, 3, 335, 6, 2, 66, 6, 3, 401, 6, 2, 72, 6, 3, 473, 6

Offset: 1

Author

Nathan Fox, Nov 23 2015

Keywords

• nonn
• easy

Comments

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) = 4, a(2) = 0, a(3) = 5, a(4) = 6, a(5) = 2, a(6) = 6, a(7) = 6, a(8) = 3.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Nathan Fox, Quasipolynomial Solutions to the Hofstadter Q-Recurrence, arXiv preprint arXiv:1511.06484 [math.NT], 2015.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).

Crossrefs

Cf. A005185, A188670, A244477, A264756, A264758.

Programs

Mathematica

Table[If[n < 3, # - n - 1, #] &@ Switch[Mod[n, 6], 0, n, 1, 6, 2, 3, 3, 3 #^2 + 3 # + 5 &[(n - 3)/6], 4, 6, 5, 2], {n, 75}] (* or *)
Rest@ CoefficientList[Series[x (4 + 5 x^2 + 6 x^3 + 2 x^4 + 6 x^5 - 6 x^6 + 3 x^7 - 4 x^8 - 12 x^9 - 4 x^10 - 6 x^11 - 6 x^13 + 5 x^14 + 6 x^15 + 2 x^16 + 2 x^18 + 3 x^19)/((1 - x)^3*(1 + x)^3*(1 - x + x^2)^3*(1 + x + x^2)^3), {x, 0, 76}], x] (* Michael De Vlieger, Nov 14 2016 *)

PARI

Vec(x*(4+5*x^2+6*x^3+2*x^4+6*x^5-6*x^6+3*x^7-4*x^8-12*x^9-4*x^10-6*x^11-6*x^13+5*x^14+6*x^15+2*x^16+2*x^18+3*x^19)/((1-x)^3*(1+x)^3*(1-x+x^2)^3*(1+x+x^2)^3) + O(x^100)) \\ Colin Barker, Nov 14 2016

Formula

a(1) = 4, a(2) = 0; thereafter a(6*n) = 6*n, a(6*n+1) = 6, a(6*n+2) = 3, a(6*n+3) = 3*n^2+3*n+5, a(6*n+4) = 6, a(6*n+5) = 2.

From Colin Barker, Nov 14 2016: (Start)

G.f.: x*(4 + 5*x^2 + 6*x^3 + 2*x^4 + 6*x^5 - 6*x^6 + 3*x^7 - 4*x^8 - 12*x^9 - 4*x^10 - 6*x^11 - 6*x^13 + 5*x^14 + 6*x^15 + 2*x^16 + 2*x^18 + 3*x^19) / ((1 - x)^3 * (1 + x)^3 * (1 - x + x^2)^3 * (1 + x + x^2)^3).

a(n) = 3*a(n-6) - 3*a(n-12) + a(n-18) for n>20.

(End)