A283881 A linear-recurrent solution to Hofstadter's Q recurrence.

7, 0, 8, 7, 7, 8, 4, 7, 7, 16, 7, 7, 16, 4, 7, 14, 24, 7, 7, 32, 4, 7, 21, 32, 7, 7, 64, 4, 7, 28, 40, 7, 7, 128, 4, 7, 35, 48, 7, 7, 256, 4, 7, 42, 56, 7, 7, 512, 4, 7, 49, 64, 7, 7, 1024, 4, 7, 56, 72, 7, 7, 2048, 4, 7, 63, 80, 7, 7, 4096, 4, 7, 70, 88, 7, 7

Offset

1,1

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

This sequence is an interleaving of seven simpler sequences. Four are constant, two are linear polynomials, and one is a geometric sequence.

Links

Nathan Fox, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 2).

Formula

a(7n) = 4, a(7n+1) = 7, a(7n+2) = 7n, a(7n+3) = 8n+8, a(7n+4) = 7, a(7n+5) = 7, a(7n+6) = 2^(n+3).

G.f.: (-8*x^20-8*x^19-14*x^18-14*x^17+14*x^15-14*x^14+12*x^13 +16*x^12 +21*x^11 +21*x^10+16*x^9-7*x^8+21*x^7-4*x^6-8*x^5-7*x^4-7*x^3 -8*x^2-7) / ((-1+2*x^7)*(-1+x)^2*(1+x+x^2+x^3+x^4+x^5+x^6)^2).

a(n) = 4*a(n-7) - 5*a(n-14) + 2*a(n-21) for n > 21.

Maple

A283881:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 7: elif n = 2 then 0: elif n = 3 then 8: elif n = 4 then 7: elif n = 5 then 7: elif n = 6 then 8: elif n = 7 then 4: else A283881(n-A283881(n-1)) + A283881(n-A283881(n-2)): fi: end:

Crossrefs

Cf. A005185, A188670, A244477, A269328, A275153, A283880.

Sequence in context: A198939 A290372 A190410 * A178308 A320377 A213186

Adjacent sequences: A283878 A283879 A283880 * A283882 A283883 A283884

Keyword

nonn

Author

Nathan Fox, Mar 19 2017

Status

approved