A063882 -id:A063882 - cp's OEIS frontend

A285762 A slow relative of Hofstadter's Q sequence.

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 12, 12, 12, 13, 14, 15, 15, 15, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 18, 18, 18, 19, 20, 21, 21, 21, 21, 22, 23, 24, 24, 24, 24, 24, 24, 24, 25, 26, 27, 27, 27, 27, 28

Offset: 1

Nathan Fox, Apr 25 2017

nonn

a(n) is the solution to the recurrence relation a(n) = a(n-12-a(n-3)) + a(n-12-a(n-12)), with a(1) through a(33) as initial conditions.
The sequence a(n) is monotonic, with successive terms increasing by 0 or 1. So the sequence hits every positive integer.
This sequence can be obtained from A285761 using a construction of Isgur et al.

Nathan Fox, Table of n, a(n) for n = 1..10000
A. Isgur, R. Lech, S. Moore, S. Tanny, Y. Verberne, and Y. Zhang, Constructing New Families of Nested Recursions with Slow Solutions, SIAM J. Discrete Math., 30(2), 2016, 1128-1147. (20 pages); DOI:10.1137/15M1040505

Cf. A005185, A063882, A285757, A285758, A285759, A285760, A285761.

A285762:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 1: elif n = 2 then 2: elif n = 3 then 3: elif n = 4 then 4: elif n = 5 then 5: elif n = 6 then 6: elif n = 7 then 7: elif n = 8 then 8: elif n = 9 then 9: elif n = 10 then 9: elif n = 11 then 9: elif n = 12 then 9: elif n = 13 then 9: elif n = 14 then 9: elif n = 15 then 9: elif n = 16 then 10: elif n = 17 then 11: elif n = 18 then 12: elif n = 19 then 12: elif n = 20 then 12: elif n = 21 then 12: elif n = 22 then 12: elif n = 23 then 12: elif n = 24 then 12: elif n = 25 then 13: elif n = 26 then 14: elif n = 27 then 15: elif n = 28 then 15: elif n = 29 then 15: elif n = 30 then 15: elif n = 31 then 15: elif n = 32 then 15: elif n = 33 then 15: else A285762(n-12-A285762(n-3)) + A285762(n-12-A285762(n-12)): fi: end:

A134680 a(n) = length (or lifetime) of the meta-Fibonacci sequence {f(1) = ... = f(n) = 1; f(k)=f(k-f(k-1))+f(k-f(k-n))} if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

Original entry on oeis.org

6, 0, 164, 0, 60, 2354, 282, 1336, 100, 1254, 366, 419, 498, 483, 778, 1204, 292, 373, 845, 838, 1118, 2120, 815, 2616, 686, 1195, 745, 1112, 2132, 1588, 754, 1227, 1279, 1661, 716, 2275, 784, 2341, 1874, 1463, 1122, 2800, 1350, 1613, 2279, 1557, 1532

Offset: 1

T. D. Noe, Nov 06 2007

nonn

Such a sequence has finite length when the k-th term becomes greater than k.

The term a(2) = 0 is only conjectural - see A005185. a(4) = 0 is a theorem of Balamohan et al. (2007). - N. J. A. Sloane, Nov 07 2007, Apr 18 2014.

			a(1) = 6: the f-sequence is defined by f(1) = 1, f(n) = 2f(n-f(n-1)), which gives 1,2,2,4,2,8 but f(7) = 2f(-1) is undefined, so the length is 6.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
D. R. Hofstadter, Curious patterns and non-patterns in a family of meta-Fibonacci recursions, Lecture in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, April 10 2014; Part 1, Part 2.
D. R. Hofstadter, Graph of first 21500 terms.
Index entries for Hofstadter-type sequences

Cf. A005185, A046700, A063882, A132172, A134679 (sequences for n=2..6).
See A240810 for another version.
A diagonal of the triangle in A240813.

Table[Clear[a]; a[n_] := a[n] = If[n<=k, 1, a[n-a[n-1]]+a[n-a[n-k]]]; t={1}; n=2; While[n<10000 && a[n-1]

A309567 a(1) = 4, a(2) = 2, a(3) = 5, a(4) = 3, a(5) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 5.

Original entry on oeis.org

4, 2, 5, 3, 1, 4, 7, 5, 8, 6, 4, 12, 5, 13, 6, 9, 17, 5, 18, 6, 9, 22, 5, 23, 11, 9, 27, 5, 28, 11, 9, 32, 5, 33, 11, 14, 37, 5, 38, 11, 14, 42, 5, 43, 11, 14, 47, 5, 48, 16, 14, 52, 5, 53, 16, 14, 57, 5, 58, 16, 14, 62, 5, 63, 16, 19, 67, 5, 68, 16, 19, 72, 5, 73, 16, 19, 77, 5, 78, 16, 19, 82, 5, 83, 21, 19, 87, 5

Offset: 1

Altug Alkan and Rémy Sigrist, Aug 08 2019

A well-defined quasi-periodic solution for Hofstadter V recurrence (a(n) = a(n-a(n-1)) + a(n-a(n-4))).

Robert Israel, Table of n, a(n) for n = 1..10000
Altug Alkan, Nathan Fox, Orhan Ozgur Aybar, Zehra Akdeniz, On Some Solutions to Hofstadter's V-Recurrence, arXiv:2002.03396 [math.DS], 2020.

Cf. A063882, A244477, A296518, A309492, A309494, A309496, A309554.

f:= proc(n) local k,j;
  j:= n mod 5;
  k:= (n-j)/5;
  if j=0 then 5*floor(sqrt(k-1))+1
  elif j=1 then 5*round(sqrt(k))-1
  elif j=2 then 5*k+2
  elif j=3 then 5
  else 5*k+3
  fi
end proc:
f(1):= 4:
map(f, [$1..100]); # Robert Israel, Aug 08 2019

a[n_] := a[n] = If[n < 6, {4, 2, 5, 3, 1}[[n]], a[n - a[n-1]] + a[n - a[n-4]]]; Array[a, 88] (* Giovanni Resta, Aug 08 2019 *)

q=vector(100); q[1]=4; q[2]=2; q[3]=5; q[4]=3; q[5]=1; for(n=6, #q, q[n]=q[n-q[n-1]]+q[n-q[n-4]]); q

For k >= 1:
a(5*k)   = 5*floor(sqrt(k-1))+1,
a(5*k+1) = 5*round(sqrt(k))-1,
a(5*k+2) = 5*k+2,
a(5*k+3) = 5,
a(5*k+4) = 5*k+3.

A240818 a(n) = length (or lifetime) of the meta-Fibonacci sequence f(k) = k for k <= n; f(k)=f(k-f(k-1))+f(k-f(k-n)) if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

Original entry on oeis.org

6, 0, 162, 0, 56, 2349, 276, 1300, 84, 1245, 356, 408, 486, 470, 764, 1172, 258, 356, 805, 819, 1078, 2099, 470, 2593, 662, 1170, 665, 1085, 2104, 1417, 724, 1196, 1247, 1628, 648, 2240, 712, 2304, 1836, 1424, 1082, 2759, 1264, 1570, 2235, 1512, 1442, 2447

Offset: 1

N. J. A. Sloane, Apr 15 2014

nonn

The terms a(2) = 0 and a(4) = 0 are only conjectural.

This sequence is very similar to A134680.

D. R. Hofstadter, Curious patterns and non-patterns in a family of meta-Fibonacci recursions, Lecture in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, April 10 2014.

Lars Blomberg, Table of n, a(n) for n = 1..10000, "infinity" = 10^8.
D. R. Hofstadter, Curious patterns and non-patterns in a family of meta-Fibonacci recursions, Lecture in Doron Zeilberger's Experimental Mathematics Seminar, Rutgers University, April 10 2014; Part 1, Part 2.
Index entries for Hofstadter-type sequences

The sequences for n=2,3,4 are A005185 and (essentially) A046700, A063882.
See A240822 for another version.
A diagonal of the triangle in A240821.
Cf. A134680.

More terms from Lars Blomberg, Oct 24 2014

A309636 a(1) = 3, a(2) = 1, a(3) = 4, a(4) = 2, a(5) = 5; a(6) = 3; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 6.

Original entry on oeis.org

3, 1, 4, 2, 5, 3, 6, 4, 7, 10, 8, 6, 9, 7, 10, 13, 6, 14, 12, 10, 18, 6, 14, 17, 10, 23, 11, 14, 22, 10, 28, 16, 14, 27, 10, 33, 16, 14, 32, 10, 38, 16, 19, 37, 10, 43, 16, 24, 42, 10, 48, 16, 24, 47, 10, 53, 16, 24, 52, 10, 58, 16, 24, 57, 10, 63, 21, 24, 62, 10, 68, 26, 24, 67, 10

Offset: 1

Altug Alkan and Nathan Fox, Aug 10 2019

A well-defined quasi-periodic solution for Hofstadter V recurrence (a(n) = a(n-a(n-1)) + a(n-a(n-4))).

Michael De Vlieger, Table of n, a(n) for n = 1..10006
Altug Alkan, Nathan Fox, Orhan Ozgur Aybar, Zehra Akdeniz, On Some Solutions to Hofstadter's V-Recurrence, arXiv:2002.03396 [math.DS], 2020.

Cf. A063882, A244477, A296518, A309492, A309494, A309496, A309554, A309567.

I:=[3,1,4,2,5,3]; [n le 6 select I[n] else  Self(n-Self(n-1)) + Self(n-Self(n-4)): n in [1..80]]; // Marius A. Burtea, Aug 11 2019

Nest[Append[#, #[[-#[[-1]] ]] + #[[-#[[-4]] ]]] &, {3, 1, 4, 2, 5, 3}, 69] (* Michael De Vlieger, May 08 2020 *)

q=vector(100); q[1]=3; q[2]=1; q[3]=4; q[4]=2; q[5]=5; q[6]=3; for(n=7, #q, q[n] = q[n-q[n-1]] + q[n-q[n-4]]); q

For k > 1:
a(5*k)   = 10,
a(5*k+1) = 5*k-2,
a(5*k+2) = 5*(floor((sqrt(2*k-1)-1)/2) + floor((sqrt(2*k-3)-1)/2)) + 6,
a(5*k+3) = 5*(floor(sqrt(k/2)) + floor(sqrt((k-1)/2))) + 4,
a(5*k+4) = 5*k-3.
Also, a(5*k+2) = 5*f(k)+1 and a(5*k+3) = 5*g(k)-1 where f(k) = g(k-g(k-1)) and g(k) = f(k-f(k))+2 with f(1) = g(1) = 1, g(2) = 2.

A296388 a(1) = a(3) = 1, a(2) = a(4) = 3, a(5) = 4; a(n) = n - a(n-a(n-1)) - a(n-a(n-4)) for n > 5.

Original entry on oeis.org

1, 3, 1, 3, 4, 2, 1, 3, 3, 6, 1, 8, 3, 10, 1, 12, 3, 14, 1, 16, 3, 18, 1, 20, 3, 22, 1, 24, 3, 26, 1, 28, 3, 30, 1, 32, 3, 34, 1, 36, 3, 38, 1, 40, 3, 42, 1, 44, 3, 46, 1, 48, 3, 50, 1, 52, 3, 54, 1, 56, 3, 58, 1, 60, 3, 62, 1, 64, 3, 66, 1, 68, 3, 70, 1, 72, 3, 74, 1, 76, 3, 78, 1, 80, 3, 82, 1, 84, 3, 86

Offset: 1

Altug Alkan, Dec 11 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A063882, A289205.

Fold[Append[#1, #2 - #1[[#2 - #1[[#2 - 1]] ]] - #1[[#2 - #1[[#2 - 4]] ]] ] &, {1, 3, 1, 3, 4}, Range[6, 90]] (* Michael De Vlieger, Dec 11 2017 *)

q=vector(10^5); q[1]=1;q[2]=3;q[3]=1;q[4]=3;q[5]=4;for(n=6, #q, q[n] = n-q[n-q[n-1]]-q[n-q[n-4]]); q

Vec(x*(1 + 3*x + 2*x^4 - 4*x^5 - 3*x^6 + x^7 - x^8 + 4*x^9 + x^10 + x^11 - x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)) + O(x^100)) \\ Colin Barker, Dec 12 2017

a(2*k) = 2*(k-2) for k >= 5, a(4*j+1) = 3 for j >= 2, a(4*m-1) = 1 for m >= 1.
From Colin Barker, Dec 12 2017: (Start)
G.f.: x*(1 + 3*x + 2*x^4 - 4*x^5 - 3*x^6 + x^7 - x^8 + 4*x^9 + x^10 + x^11 - x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)).
a(n) = a(n-2) + a(n-4) - a(n-6) for n>12.
(End)

A304590 a(1) = a(2) = a(3) = a(5) = 1, a(4) = 2, a(6) = 3; a(n) = a(n-a(n-3)) + a(n-a(n-6)) for n > 6.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 4, 8, 11, 11, 12, 12, 12, 4, 3, 3, 13, 6, 7, 7, 9, 10, 14, 9, 9, 19, 12, 13, 14, 15, 16, 18, 15, 15, 22, 18, 19, 10, 18, 18, 23, 18, 18, 24, 24, 26, 23, 24, 30, 31, 24, 29, 25, 30, 28, 25, 27, 29, 30, 30, 33, 37, 33, 38, 35, 33, 29, 31, 36, 41, 36, 36, 42, 28, 36, 37, 53, 36, 37, 41, 48, 48, 33

Offset: 1

Altug Alkan, May 15 2018

nonn

Altug Alkan, Table of n, a(n) for n = 1..10000

Cf. A005185, A063882, A087777, A240827.

a:=[1,1,1,2,1,3];; for n in [7..100] do a[n]:=a[n-a[n-3]]+a[n-a[n-6]]; od; a; # Muniru A Asiru, May 19 2018

f:= proc(n) option remember; procname(n-procname(n-3))+procname(n-procname(n-6)) end proc:
for i from 1 to 6 do f(i):= [1,1,1,2,1,3][i] od:
map(f, [$1..100]); # Robert Israel, May 16 2018

Nest[Append[#2, #2[[#1 - #2[[-3]] ]] + #2[[#1 - #2[[-6]] ]] ] & @@ {Length@ # + 1, #} &, {1, 1, 1, 2, 1, 3}, 77] (* Michael De Vlieger, Jul 20 2018 *)

q=vector(100); q[1]=q[2]=q[3]=q[5]=1;q[4]=2;q[6]=3; for(n=7, #q, q[n]=q[n-q[n-3]]+q[n-q[n-6]]); q

A309704 a(1) = 3, a(2) = 4, a(3) = 5, a(4) = 4, a(5) = 5; a(6) = 6; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 6.

Original entry on oeis.org

3, 4, 5, 4, 5, 6, 7, 7, 8, 8, 9, 10, 10, 10, 11, 11, 13, 12, 14, 14, 14, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 24, 23, 23, 25, 24, 26, 27, 27, 26, 28, 28, 28, 29, 29, 30, 31, 31, 32, 32, 33, 33, 34, 35, 35, 35, 36, 36, 37, 37, 38, 39, 39, 39, 40, 40, 40, 41, 41, 42, 43, 43, 45, 45, 45, 45, 48, 44, 48, 49, 47, 52, 47, 51, 50, 47, 52, 50, 54, 52, 54, 55, 54, 54, 56

Offset: 1

Altug Alkan and Rémy Sigrist, Aug 13 2019

This sequence is finite but has an exceptionally long life: a(3080193026) = 3101399868 is its last term since a(3080193027) refers to a nonpositive index and thus fails to exist. See plots in Links section to fractal-like structure of a(n)-n/2.

Altug Alkan, Table of n, a(n) for n = 1..20000
Altug Alkan, Nathan Fox, Orhan Ozgur Aybar, Zehra Akdeniz, On Some Solutions to Hofstadter's V-Recurrence, arXiv:2002.03396 [math.DS], 2020.
Rémy Sigrist, Density plot of a(n)-n/2 for n <= 10^8
Rémy Sigrist, Scatterplot of a(n)-n/2 for n <= 10^8
Rémy Sigrist, Density plot of the whole sequence (n = 1..3080193026)

Cf. A005185, A063882, A309567, A309636.

Nest[Append[#, #[[-#[[-1]] ]] + #[[-#[[-4]] ]]] &, {3, 4, 5, 4, 5, 6}, 94] (* Michael De Vlieger, May 08 2020 *)

q=vector(100); q[1]=3; q[2]=4; q[3]=5; q[4]=4; q[5]=5; q[6]=6; for(n=7, #q, q[n] = q[n-q[n-1]] + q[n-q[n-4]]); q

a(3080193026) from Giovanni Resta, Aug 13 2019

A309967 a(1) = a(2) = 1, a(3) = 2, a(4) = 3, a(5) = 8, a(6) = 6, a(7) = a(8) = 4; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 8.

Original entry on oeis.org

1, 1, 2, 3, 8, 6, 4, 4, 9, 4, 8, 7, 9, 12, 6, 13, 7, 14, 17, 6, 18, 7, 19, 22, 6, 23, 7, 24, 27, 6, 28, 7, 29, 32, 6, 33, 7, 34, 37, 6, 38, 7, 39, 42, 6, 43, 7, 44, 47, 6, 48, 7, 49, 52, 6, 53, 7, 54, 57, 6, 58, 7, 59, 62, 6, 63, 7, 64, 67, 6, 68, 7, 69, 72, 6, 73, 7, 74, 77, 6, 78, 7

Offset: 1

Altug Alkan, Aug 25 2019

A quasilinear solution sequence for Hofstadter V recurrence (a(n) = a(n-a(n-1)) + a(n-a(n-4))).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. A005185, A063882, A244477, A309567, A309636.

q=vector(100); q[1]=q[2]=1; q[3]=2; q[4]=3; q[5]=8; q[6]=6; q[7]=q[8]=4; for(n=9, #q, q[n]=q[n-q[n-1]]+q[n-q[n-4]]); q

Vec(x*(1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 4*x^5 + 2*x^6 + 3*x^8 - 12*x^9 - 3*x^10 +  3*x^12 - 3*x^13 + 6*x^14 + 3*x^15 - 3*x^16 + 2*x^18 - 2*x^19) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)^2) + O(x^40)) \\ Colin Barker, Aug 25 2019

For k > 2:
a(5*k-4) = 5*k-7,
a(5*k-3) = 7,
a(5*k-2) = 5*k-6,
a(5*k-1) = 5*k-3,
a(5*k) = 6.
From Colin Barker, Aug 25 2019: (Start)
G.f.: x*(1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 4*x^5 + 2*x^6 + 3*x^8 - 12*x^9 - 3*x^10 +  3*x^12 - 3*x^13 + 6*x^14 + 3*x^15 - 3*x^16 + 2*x^18 - 2*x^19) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)^2).
a(n) = 2*a(n-5) - a(n-10) for n > 20.
(End)

A309430 a(n) = a(n-a(n-1)) + a(n-a(n-4)), with a(n) = ceiling(2*n/3) for n <= 7.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 11, 12, 13, 13, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 31, 32, 33, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 38, 38, 39, 39, 40, 41, 41, 41, 42, 42, 43, 44, 44

Offset: 1

Altug Alkan, Aug 01 2019

A slow solution to Hofstadter V recurrence.

Numbers k such that a(k) < A063882(k) are 1654, 1721, 1925, ...

Cf. A063882, A317686, A319020.

q=vector(100); for(n=1, 7, q[n]=ceil(2*n/3));for(n=8, #q, q[n]=q[n-q[n-1]] +q[n-q[n-4]]); q

a(n+1) - a(n) = 0 or 1 for all n >= 1.

cp's OEIS Frontend

A285762 A slow relative of Hofstadter's Q sequence.

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Maple

A134680 a(n) = length (or lifetime) of the meta-Fibonacci sequence {f(1) = ... = f(n) = 1; f(k)=f(k-f(k-1))+f(k-f(k-n))} if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

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A309567 a(1) = 4, a(2) = 2, a(3) = 5, a(4) = 3, a(5) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 5.

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A240818 a(n) = length (or lifetime) of the meta-Fibonacci sequence f(k) = k for k <= n; f(k)=f(k-f(k-1))+f(k-f(k-n)) if that sequence is only defined for finitely many terms, or 0 if that sequence is infinite.

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A309636 a(1) = 3, a(2) = 1, a(3) = 4, a(4) = 2, a(5) = 5; a(6) = 3; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 6.

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Magma

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A296388 a(1) = a(3) = 1, a(2) = a(4) = 3, a(5) = 4; a(n) = n - a(n-a(n-1)) - a(n-a(n-4)) for n > 5.

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A304590 a(1) = a(2) = a(3) = a(5) = 1, a(4) = 2, a(6) = 3; a(n) = a(n-a(n-3)) + a(n-a(n-6)) for n > 6.

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A309704 a(1) = 3, a(2) = 4, a(3) = 5, a(4) = 4, a(5) = 5; a(6) = 6; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 6.

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