A283904 Relative of Hofstadter Q-sequence: a(1) = 1, a(2) = 1; thereafter a(n) = a(n-2a(n-1)) + a(n-3a(n-2)).

1, 1, 1, 2, 2, 1, 3, 3, 1, 4, 4, 2, 2, 7, 2, 2, 6, 3, 3, 11, 2, 3, 12, 2, 2, 14, 2, 2, 14, 3, 2, 14, 4, 14, 15, 1, 15, 18, 1, 18, 20, 1, 20, 22, 1, 22, 23, 1, 23, 25, 1, 25, 26, 1, 26, 28, 1, 28, 29, 1, 29, 31, 1, 31, 32, 1, 32, 34, 1, 34, 35, 1, 35, 37, 1

Offset: 1

Author

Nathan Fox, Mar 19 2017

Keywords

• nonn

Comments

This sequence is eventually quasilinear with period 6. Each component sequence has slope 0 or 1/2.

Links

• Nathan Fox, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0, 0, 1, 0, 0, 1, 0, 0, -1).

Crossrefs

Cf. A005185, A188670, A244477.

Programs

Maple

A283904:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 1: elif n = 2 then 1: else A283904(n-2*A283904(n-1)) + A283904(n-3*A283904(n-2)): fi: end:

Formula

If the index is at least 41:

a(6n) = 1

a(6n+1) = 3n-1

a(6n+2) = 3n+1

a(6n+3) = 1

a(6n+4) = 3n+1

a(6n+5) = 3n+2.

G.f.: (-x^48-x^46-x^45-x^43+x^42+3*x^41+x^40-9*x^39-x^38+3*x^37+x^36-4*x^35+x^34+12*x^33+x^32-2*x^31+x^30+x^29-x^28+x^26-3*x^25-2*x^24-x^23+2*x^22-x^20+2*x^19+3*x^18-2*x^16-x^15+2*x^13-3*x^12+x^11-x^8+x^4+x^3+x^2+x+1) / ((1+x)*(1-x+x^2)*(-1+x)^2*(1+x+x^2)^2).

a(n) = a(n-3) + a(n-6) - a(n-9) for n > 49.