A264756 -id:A264756 - cp's OEIS frontend

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

Nathan Fox, Nov 23 2015

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.

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).

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

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 *)

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

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)

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

Original entry on oeis.org

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, 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, 3, 542, 9, 2, 63, 9, 3, 257, 9, 3, 799, 9, 2, 72, 9, 3, 329, 9

Offset: 1

Nathan Fox, Nov 23 2015

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.

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 (0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,-1).

Cf. A005185, A188670, A244477, A264756, A264757.

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 *)

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

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.
From Colin Barker, Nov 14 2016: (Start)
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).
a(n) = 4*a(n-9) - 6*a(n-18) + 4*a(n-27) - a(n-36) for n>38.
(End)

A275153 A linear-recurrent solution to Hofstadter's Q-recurrence.

Original entry on oeis.org

9, 0, 0, 0, 7, 9, 9, 10, 4, 9, 9, 3, 9, 16, 9, 9, 20, 4, 9, 18, 3, 9, 34, 9, 9, 40, 4, 9, 27, 3, 9, 61, 9, 9, 80, 4, 9, 36, 3, 9, 97, 9, 9, 160, 4, 9, 45, 3, 9, 142, 9, 9, 320, 4, 9, 54, 3, 9, 196, 9, 9, 640, 4, 9, 63, 3, 9, 259, 9, 9, 1280, 4, 9, 72, 3, 9, 331, 9, 9, 2560

Offset: 1

Nathan Fox, Jul 17 2016

nonn

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) = 9, a(2) = 0, a(3) = 0, a(4) = 0, a(5) = 7, a(6) = 9, a(7) = 9, a(8) = 10, a(9) = 4, a(10) = 9, a(11) = 9, a(12) = 3.

This sequence is an interleaving of nine simpler sequences. Six are eventually constant, one is a linear polynomial, one is a quadratic polynomial, and one is a geometric sequence.

Nathan Fox, Table of n, a(n) for n = 1..1000
Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, -2).

Cf. A005185, A188670, A244477, A264756, A264757, A264758, A268368.

Join[{9, 0, 0, 0}, LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, -2}, {7, 9, 9, 10, 4, 9, 9, 3, 9, 16, 9, 9, 20, 4, 9, 18, 3, 9, 34, 9, 9, 40, 4, 9, 27, 3, 9, 61, 9, 9, 80, 4, 9, 36, 3, 9}, 100]] (* Jean-François Alcover, Dec 01 2018 *)

a(3) = 0, a(4) = 0; otherwise:
a(9n) = 4,     a(9n+1) = 9         a(9n+2) = 9n      a(9n+3) = 3.
a(9n+4) = 9,   a(9n+5) = (9*n^2 + 9*n + 14)/2,     a(9n+6) = 9.
a(9n+7) = 9,   a(9n+8) = 10*2^n.
a(n) = 5*a(n-9) - 9*a(n-18) + 7*a(n-27) - 2*a(n-36) for n>40.
G.f.: -(18*x^39 +6*x^38 +8*x^35 +10*x^34 +18*x^33 +18*x^32 +14*x^31 -45*x^30 -15*x^29 -18*x^28 +18*x^27 -20*x^26 -30*x^25 -45*x^24 -45*x^23 -17*x^22 +36*x^21 +12*x^20 +27*x^19 -45*x^18 +16*x^17 +30*x^16 +36*x^15 +36*x^14 +19*x^13 -9*x^12 -3*x^11 -9*x^10 +36*x^9 -4*x^8 -10*x^7 -9*x^6 -9*x^5 -7*x^4 -9)/((2*x^9-1)*(x-1)^3*(x^2+x+1)^3*(x^6+x^3+1)^3).

A269328 An eventually quasilinear solution to Hofstadter's Q recurrence.

Original entry on oeis.org

5, 2, 0, 3, 6, 5, 2, 5, 5, 12, 5, 2, 10, 5, 18, 5, 2, 15, 5, 24, 5, 2, 20, 5, 30, 5, 2, 25, 5, 36, 5, 2, 30, 5, 42, 5, 2, 35, 5, 48, 5, 2, 40, 5, 54, 5, 2, 45, 5, 60, 5, 2, 50, 5, 66, 5, 2, 55, 5, 72, 5, 2, 60, 5, 78, 5, 2, 65, 5, 84, 5, 2, 70, 5, 90

Offset: 1

Nathan Fox, Feb 23 2016

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) = 5, a(2) = 2, a(3) = 0, a(4) = 3, a(5) = 6, a(6) = 5, a(7) = 2.
Starting from n=5, this sequence consists of five interleaved linear sequences with three different slopes.
Square array read by rows: T(j,k), j>=1, 1<=k<=5, in which row j list [5, 2, 5*(j-1), 5, 6*j], except T(1,4) = 3, not 5. - Omar E. Pol, Jun 22 2016

			From _Omar E. Pol_, Jun 22 2016: (Start)
Written as a square array T(j,k) with five columns the sequence begins:
5, 2,  0, 3,  6;
5, 2,  5, 5, 12;
5, 2, 10, 5, 18;
5, 2, 15, 5, 24;
5, 2, 20, 5, 30;
5, 2, 25, 5, 36;
5, 2, 30, 5, 42;
5, 2, 35, 5, 48;
5, 2, 40, 5, 54;
5, 2, 45, 5, 60;
5, 2, 50, 5, 66;
5, 2, 55, 5, 72;
5, 2, 60, 5, 78;
5, 2, 65, 5, 84;
5, 2, 70, 5, 90;
...
Note that T(1,4) = 3, not 5. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 2, 0, 0, 0, 0, -1).

Cf. A005185, A188670, A244477, A264756.

I:=[5,2,0,3,6,5,2,5,5,12,5,2,10,5]; [n le 14 select I[n] else 2*Self(n-5)-Self(n-10): n in [1..100]]; // Vincenzo Librandi, Dec 16 2018

Join[{5, 2, 0, 3}, LinearRecurrence[{0, 0, 0, 0, 2, 0, 0, 0, 0, -1} , {6, 5, 2, 5, 5, 12, 5, 2, 10, 5}, 80]] (* Jean-François Alcover, Dec 16 2018 *)
CoefficientList[Series[(-2 x^13 - x^8 + 5 x^7 - 2 x^6 - 5 x^5 + 6 x^4 + 3 x^3 + 2 x + 5) / (x^10 - 2 x^5 + 1), {x, 0, 100}], x] (* Vincenzo Librandi, Dec 16 2018 *)

a(4) = 3; otherwise a(5n) = 6n, a(5n+1) = 5, a(5n+2) = 2, a(5n+3) = 5n, a(5n+4) = 5.
From Chai Wah Wu, Jun 22 2016: (Start)
a(n) = 2*a(n-5) - a(n-10) for n > 14.
G.f.: x*(-2*x^13 - x^8 + 5*x^7 - 2*x^6 - 5*x^5 + 6*x^4 + 3*x^3 + 2*x + 5)/(x^10 - 2*x^5 + 1). (End)

A275361 An eventually quasilinear solution to Hofstadter's Q-recurrence.

Original entry on oeis.org

0, 4, -40, -9, 8, -8, 7, 1, 5, 13, -24, -1, 8, 8, 8, 1, 5, 13, -8, 7, 8, 8, 23, 1, 5, 13, 8, 15, 8, 16, 31, 1, 5, 13, 24, 23, 8, 24, 39, 1, 5, 13, 40, 31, 8, 32, 47, 1, 5, 13, 56, 39, 8, 40, 55, 1, 5, 13, 72, 47, 8, 48, 63, 1, 5, 13, 88, 55, 8, 56, 71

Offset: 1

Nathan Fox, Jul 24 2016

sign

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 first 45 terms as initial conditions.

This is a quasilinear sequence with quasiperiod 8. Four of the component sequences are constant, three have slope 1, and one has slope 2.

Nathan Fox, Table of n, a(n) for n = 1..1000
Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A005185, A188670, A244477, A264756, A264757, A264758, A268368, A275153, A275362.

Join[{0, 4, -40, -9, 8, -8, 7, 1, 5, 13, -24, -1, 8, 8, 8}, LinearRecurrence[ {0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -1}, {1, 5, 13, -8, 7, 8, 8, 23, 1, 5, 13, 8, 15, 8, 16, 31}, 100]] (* Jean-François Alcover, Dec 12 2018 *)

a(1) = 0, a(2) = 4, a(14) = 8, a(15) = 8; otherwise:
a(8n) = 1, a(8n+1) = 5, a(8n+2) = 13, a(8n+3) = 16n-40, a(8n+4) = 8n-9, a(8n+5) = 8, a(8n+6) = 8n-8, a(8n+7) = 8n+7.
a(n) = 2*a(n-8) - a(n-16) for n>31.
G.f.: -(7*x^30 -8*x^29 -14*x^22 +16*x^21 +9*x^17 +5*x^16 +x^15 +6*x^14 -24*x^13 +8*x^12 -17*x^11 -56*x^10 -5*x^9 -5*x^8 -x^7 -7*x^6 +8*x^5 -8*x^4 +9*x^3 +40*x^2 -4*x)/((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)^2).

A275362 An eventually quasilinear solution to Hofstadter's Q recurrence.

Original entry on oeis.org

-9, 2, 9, 2, 0, 7, 9, 10, 3, 0, 2, 9, 2, 9, 9, 9, 20, 3, 9, 22, 9, 2, 18, 9, 18, 30, 3, 18, 32, 9, 2, 27, 9, 27, 40, 3, 27, 42, 9, 2, 36, 9, 36, 50, 3, 36, 52, 9, 2, 45, 9, 45, 60, 3, 45, 62, 9, 2, 54, 9, 54, 70, 3, 54, 72, 9, 2, 63, 9, 63, 80, 3, 63, 82

Offset: 1

Nathan Fox, Jul 24 2016

sign

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) = -9, a(2) = 2, a(3) = 9, a(4) = 2, a(5) = 0, a(6) = 7, a(7) = 9, a(8) = 10, a(9) = 3, a(10) = 0, a(11) = 2, a(12) = 9, a(13) = 2, a(14) = 9, a(15) = 9, a(16) = 9.

This is a quasilinear sequence with quasiperiod 9. Four of the component sequences are constant, three have slope 1, and two have slope 10/9.

Nathan Fox, Table of n, a(n) for n = 1..1000
Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A005185, A188670, A244477, A264756, A264757, A264758, A268368, A275153, A275361.

Join[{-9, 2, 9, 2, 0, 7, 9, 10, 3, 0, 2}, LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1}, {9, 2, 9, 9, 9, 20, 3, 9, 22, 9, 2, 18, 9, 18, 30, 3, 18, 32}, 100]] (* Jean-François Alcover, Dec 13 2018 *)

a(6) = 7, a(7) = 9, a(11) = 2; otherwise:
a(9n) = 3, a(9n+1) = 9n-9, a(9n+2) = 10n+2, a(9n+3) = 9, a(9n+4) = 2, a(9n+5) = 9n, a(9n+6) = 9, a(9n+7) = 9n, a(9n+8) = 10n+10.
a(n) = 2*a(n-9) - a(n-18) for n>29.
G.f.: -(10*x^28 -9*x^24 +2*x^23 -20*x^19 +3*x^17 +9*x^15 +5*x^14 -9*x^13 +2*x^12 +9*x^11 +2*x^10 -18*x^9 -3*x^8 -10*x^7 -9*x^6 -7*x^5 -2*x^3 -9*x^2 -2*x+9)/((x-1)^2*(x^2+x+1)^2*(x^6+x^3+1)^2).

A283878 An eventually quasilinear solution to Hofstadter's Q recurrence.

Original entry on oeis.org

0, 2, 3, 1, 3, 6, 1, 3, 9, 1, 3, 12, 1, 3, 15, 1, 3, 18, 1, 3, 21, 1, 3, 24, 1, 3, 27, 1, 3, 30, 1, 3, 33, 1, 3, 36, 1, 3, 39, 1, 3, 42, 1, 3, 45, 1, 3, 48, 1, 3, 51, 1, 3, 54, 1, 3, 57, 1, 3, 60, 1, 3, 63, 1, 3, 66, 1, 3, 69, 1, 3, 72, 1, 3, 75

Offset: 1

Nathan Fox, Mar 19 2017

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) = 0, a(2) = 2, a(3) = 3, a(4) = 1.

Nathan Fox, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 2, 0, 0, -1).

Cf. A005185, A188670, A244477, A264756, A283879.

A283878:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 0: elif n = 2 then 2: elif n = 3 then 3: elif n = 4 then 1: else A283878(n-A283878(n-1)) + A283878(n-A283878(n-2)): fi: end:

G.f.: (-x*(x^6 + x^5 + x^3 - x^2 - 3*x - 2)) / ((-1 + x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-3) - a(n-6) for n > 8.
a(1) = 0, a(2) = 2; thereafter a(3*k) = 3*k, a(3*k+1) = 1, a(3*k+2) = 3.

A283879 An eventually quasilinear solution to Hofstadter's Q recurrence.

Original entry on oeis.org

4, 1, 0, 3, 3, 1, 4, 8, 7, 1, 4, 12, 11, 1, 4, 16, 15, 1, 4, 20, 19, 1, 4, 24, 23, 1, 4, 28, 27, 1, 4, 32, 31, 1, 4, 36, 35, 1, 4, 40, 39, 1, 4, 44, 43, 1, 4, 48, 47, 1, 4, 52, 51, 1, 4, 56, 55, 1, 4, 60, 59, 1, 4, 64, 63, 1, 4, 68, 67, 1, 4, 72, 71, 1, 4

Offset: 1

Nathan Fox, Mar 19 2017

nonn

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) = 1, a(3) = 0, a(4) = 3, a(5) = 3, a(6) = 1, a(7) = 4, a(8) = 8.

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

Cf. A005185, A188670, A244477, A264756, A283878.

A283879:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 4: elif n = 2 then 1: elif n = 3 then 0: elif n = 4 then 3: elif n = 5 then 3: elif n = 6 then 1: elif n = 7 then 4: elif n = 8 then 8: else A283879(n-A283879(n-1)) + A283879(n-A283879(n-2)): fi: end:

LinearRecurrence[{1,-1,1,1,-1,1,-1},{4,1,0,3,3,1,4,8,7,1,4},80] (* Harvey P. Dale, May 25 2025 *)

a(1) = 4, a(4) = 3; otherwise a(4n) = 4n, a(4n+1) = 4n-1, a(4n+2) = 1, a(4n+3) = 4.
G.f.: (-x^10-3*x^9+3*x^8+2*x^7+4*x^5-5*x^4+3*x^2-3*x+4) / ((1+x)*(-1+x)^2*(1+x^2)^2).
a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4) - a(n-5) + a(n-6) - a(n-7) for n > 11.

A275365 a(1)=2, a(2)=2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

Original entry on oeis.org

0, 2, 2, 4, 2, 6, 2, 8, 2, 10, 2, 12, 2, 14, 2, 16, 2, 18, 2, 20, 2, 22, 2, 24, 2, 26, 2, 28, 2, 30, 2, 32, 2, 34, 2, 36, 2, 38, 2, 40, 2, 42, 2, 44, 2, 46, 2, 48, 2, 50, 2, 52, 2, 54, 2, 56, 2, 58, 2, 60, 2, 62, 2, 64, 2, 66, 2, 68, 2, 70, 2, 72, 2, 74

Offset: 0

Nathan Fox, Jul 24 2016

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) = 2, a(2) = 2.
Starting with n=1, a(n) is A005843 interleaved with A007395.
This sequence is the same as A133265 with the leading 2 changed to a 0.

Nathan Fox, Table of n, a(n) for n = 0..1000
Nathan Fox, Finding Linear-Recurrent Solutions to Hofstadter-Like Recurrences Using Symbolic Computation, arXiv:1609.06342 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A005185, A133265, A188670, A244477, A264756, A264757, A264758, A268368, A275153, A275361, A275362.

Join[{0}, LinearRecurrence[{0, 2, 0, -1}, {2, 2, 4, 2}, 73]] (* Jean-François Alcover, Feb 19 2019 *)

a(0) = 0; thereafter, a(2n) = 2, a(2n+1) = 2n+2.
a(n) = 2*a(n-2) - a(n-4) for n>4.
G.f.: -(2*x^3 -2*x -2)/((x-1)^2*(x+1)^2).

A284429 A quasilinear solution to Hofstadter's Q recurrence.

Original entry on oeis.org

2, 1, 3, 5, 1, 3, 8, 1, 3, 11, 1, 3, 14, 1, 3, 17, 1, 3, 20, 1, 3, 23, 1, 3, 26, 1, 3, 29, 1, 3, 32, 1, 3, 35, 1, 3, 38, 1, 3, 41, 1, 3, 44, 1, 3, 47, 1, 3, 50, 1, 3, 53, 1, 3, 56, 1, 3, 59, 1, 3, 62, 1, 3, 65, 1, 3, 68, 1, 3, 71, 1, 3, 74, 1, 3

Offset: 1

Nathan Fox, Mar 26 2017

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(1) = 2, a(2) = 1.

This sequence is a close relative of A283878.

Nathan Fox, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 2, 0, 0, -1).

Cf. A005185, A188670, A244477, A264756, A283878, A283879.

A284429:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 2: elif n = 2 then 1: else A284429(n-A284429(n-1)) + A284429(n-A284429(n-2)): fi: end:

CoefficientList[Series[(-3*x^5 - x^4 + x^3 + 3*x^2 + x + 2) / ((-1 + x)^2*(1 + x + x^2)^2), {x, 0, 100}], x] (* Indranil Ghosh, Mar 27 2017 *)

Vec((-3*x^5 - x^4 + x^3 + 3*x^2 + x + 2) / ((-1 + x)^2*(1 + x + x^2)^2) + O(x^100)) \\ Indranil Ghosh, Mar 27 2017

G.f.: (-3*x^5 - x^4 + x^3 + 3*x^2 + x + 2) / ((-1 + x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-3) - a(n-6) for n > 6.
a(3*k)   = 3,
a(3*k+1) = 3*k+2,
a(3*k+2) = 1.

cp's OEIS Frontend

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

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

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A275153 A linear-recurrent solution to Hofstadter's Q-recurrence.

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A269328 An eventually quasilinear solution to Hofstadter's Q recurrence.

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A275361 An eventually quasilinear solution to Hofstadter's Q-recurrence.

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A275362 An eventually quasilinear solution to Hofstadter's Q recurrence.

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A283878 An eventually quasilinear solution to Hofstadter's Q recurrence.

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A283879 An eventually quasilinear solution to Hofstadter's Q recurrence.

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