A296518 -id:A296518 - cp's OEIS frontend

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.

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.

A309554 a(1) = a(6) = 1, a(2) = a(3) = a(8) = 2, a(4) = a(7) = 7, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-3)) for n > 8.

Original entry on oeis.org

1, 2, 2, 7, 5, 1, 7, 2, 9, 3, 11, 3, 6, 4, 14, 5, 9, 16, 6, 15, 6, 10, 8, 21, 21, 21, 2, 28, 3, 30, 3, 6, 4, 33, 5, 9, 35, 6, 34, 6, 10, 8, 40, 40, 40, 2, 47, 3, 49, 3, 6, 4, 52, 5, 9, 54, 6, 53, 6, 10, 8, 59, 59, 59, 2, 66, 3, 68, 3, 6, 4, 71, 5, 9, 73, 6, 72, 6, 10, 8, 78, 78, 78, 2, 85, 3, 87, 3, 6, 4

Offset: 1

Altug Alkan and Rémy Sigrist, Aug 07 2019

A well-defined solution sequence for recurrence a(n) = a(n-a(n-1)) + a(n-a(n-3)).

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

Cf. A046700, A244477, A296518, A309492, A309494, A309496.

I:=[1,2,2,7,5,1,7,2];[n le 8 select I[n] else Self(n-Self(n-1))+Self(n-Self(n-3)): n in [1..90]]; // Marius A. Burtea, Aug 08 2019

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

Vec(x*(1 + x)*(1 + x + x^2 + 6*x^3 - x^4 + 2*x^5 + 5*x^6 - 3*x^7 + 12*x^8 - 9*x^9 + 20*x^10 - 17*x^11 + 23*x^12 - 19*x^13 + 33*x^14 - 28*x^15 + 37*x^16 - 21*x^17 + 27*x^18 - 14*x^19 + 16*x^20 - 10*x^21 + 4*x^22 + 7*x^23 + 12*x^24 - 5*x^25 + 3*x^26 + 7*x^27 - 10*x^28 + 18*x^29 - 21*x^30 + 15*x^31 - 19*x^32 + 24*x^33 - 29*x^34 + 20*x^35 - 17*x^36 + 11*x^37 - 6*x^38 + 2*x^39 - 10*x^40 + 9*x^41 - 6*x^42 + 5*x^43) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18)^2) + O(x^90)) \\ Colin Barker, Aug 11 2019

For k >= 1:
a(19*k-11) = 2,
a(19*k-10) = 19*k-10,
a(19*k-9) = 3,
a(19*k-8) = 19*k-8,
a(19*k-7) = 3,
a(19*k-6) = 6,
a(19*k-5) = 4,
a(19*k-4) = 19*k-5,
a(19*k-3) = 5,
a(19*k-2) = 9,
a(19*k-1) = 19*k-3,
a(19*k) = 6,
a(19*k+1) = 19*k-4,
a(19*k+2) = 6,
a(19*k+3) = 10,
a(19*k+4) = 8,
a(19*k+5) = a(19*k+6) = a(19*k+7) = 19*k+2.
From Colin Barker, Aug 08 2019: (Start)
G.f.: x*(1 + x)*(1 + x + x^2 + 6*x^3 - x^4 + 2*x^5 + 5*x^6 - 3*x^7 + 12*x^8 - 9*x^9 + 20*x^10 - 17*x^11 + 23*x^12 - 19*x^13 + 33*x^14 - 28*x^15 + 37*x^16 - 21*x^17 + 27*x^18 - 14*x^19 + 16*x^20 - 10*x^21 + 4*x^22 + 7*x^23 + 12*x^24 - 5*x^25 + 3*x^26 + 7*x^27 - 10*x^28 + 18*x^29 - 21*x^30 + 15*x^31 - 19*x^32 + 24*x^33 - 29*x^34 + 20*x^35 - 17*x^36 + 11*x^37 - 6*x^38 + 2*x^39 - 10*x^40 + 9*x^41 - 6*x^42 + 5*x^43) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18)^2).
a(n) = 2*a(n-19) - a(n-38) for n > 45.
(End)

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.

A296690 a(1) = a(2) = a(3) = 2, a(4) = 4, a(5) = a(6) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 6.

Original entry on oeis.org

2, 2, 2, 4, 5, 5, 6, 6, 8, 10, 9, 8, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 20, 22, 21, 20, 24, 25, 20, 25, 28, 28, 21, 29, 32, 30, 25, 30, 30, 36, 30, 32, 35, 34, 36, 35, 37, 37, 38, 38, 40, 40, 40, 42, 41, 43, 43, 43, 45, 44, 46, 46, 47, 47, 49, 49, 49, 51, 50, 52, 52, 52, 54, 53, 56, 55, 58, 55, 62, 55

Offset: 1

Altug Alkan, Dec 18 2017

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Scatterplot of a(n) for n <= 500000

Cf. A005185, A278055, A284644, A292351, A296413, A296421, A296440, A296518.

a:= proc(n) option remember; procname(n-procname(n-1))+procname(n-procname(n-2))+procname(n-procname(n-3)) end proc:
a(1):= 2: a(2):= 2: a(3):= 2: a(4):= 4: a(5):= 5: a(6):= 5:
map(a, [$1..100]); #  after Robert Israel at A296440

a[n_] := a[n] = If[n<7, {2, 2, 2, 4, 5, 5}[[n]], a[n - a[n-1]] + a[n - a[n-2]] + a[n - a[n-3]]]; Array[a, 100] (* after Giovanni Resta at A296440 *)

q=vector(10^5); q[1]=2; q[2]=2; q[3]=2; q[4]=4; q[5]=5; q[6]=5; for(n=7, #q, q[n] = q[n-q[n-1]]+q[n-q[n-2]]+q[n-q[n-3]]); q

(definec (A296690 n) (cond ((<= n 3) 2) ((<= n 5) n) ((= n 6) 5) (else (+ (A296690 (- n (A296690 (- n 1)))) (A296690 (- n (A296690 (- n 2)))) (A296690 (- n (A296690 (- n 3)))))))) ;; Antti Karttunen, Dec 18 2017

A296786 a(1) = a(2) = a(5) = 2, a(3) = 1, a(4) = 3, a(6) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 6.

Original entry on oeis.org

2, 2, 1, 3, 2, 5, 7, 8, 7, 4, 11, 12, 11, 4, 15, 16, 15, 4, 19, 20, 19, 4, 23, 24, 23, 4, 27, 28, 27, 4, 31, 32, 31, 4, 35, 36, 35, 4, 39, 40, 39, 4, 43, 44, 43, 4, 47, 48, 47, 4, 51, 52, 51, 4, 55, 56, 55, 4, 59, 60, 59, 4, 63, 64, 63, 4, 67, 68, 67, 4, 71, 72, 71, 4, 75, 76, 75, 4, 79, 80, 79, 4

Offset: 1

Altug Alkan, Dec 20 2017

A quasi-periodic solution to the three-term Hofstadter recurrence a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)). See comments in A296518.

Colin Barker, Table of n, a(n) for n = 1..1000
Altug Alkan, On a conjecture about generalized Q-recurrence, Open Mathematics (2018) Vol. 16, Issue 1, 1490-1500.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A005185, A244477, A268368, A278055, A296413, A296440, A296518, A296690.

a:= proc(n) option remember; procname(n-procname(n-1))+procname(n-procname(n-2))+procname(n-procname(n-3)) end proc:
a(1):= 2: a(2):= 2: a(3):= 1: a(4):= 3: a(5):= 2: a(6):= 5:
map(a, [$1..100]); # after Robert Israel at A296440

a[n_] := a[n] = If[n<7, {2, 2, 1, 3, 2, 5}[[n]], a[n - a[n-1]] + a[n - a[n-2]] + a[n - a[n-3]]]; Array[a, 100] (* after Giovanni Resta at A296440 *)

q=vector(10^5); q[1]=2;q[2]=2;q[3]=1;q[4]=3;q[5]=2;q[6]=5;for(n=7, #q, q[n] = q[n-q[n-1]]+q[n-q[n-2]]+q[n-q[n-3]]); q

Vec(x*(2 + 2*x + x^2 + 3*x^3 - 2*x^4 + x^5 + 5*x^6 + 2*x^7 + 5*x^8 - 4*x^9 - 2*x^10 - x^11 - x^12 + x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2) + O(x^100)) \\ Colin Barker, Dec 29 2017

a(4*k-1) = a(4*k+1) = 4*k-1, a(4*k) = 4*k, a(4*k+2) = 4, for k > 1.
From Colin Barker, Dec 28 2017: (Start)
G.f.: x*(2 + 2*x + x^2 + 3*x^3 - 2*x^4 + x^5 + 5*x^6 + 2*x^7 + 5*x^8 - 4*x^9 - 2*x^10 - x^11 - x^12 + x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
a(n) = 2*a(n-4) - a(n-8) for n>14.
(End)

A304490 a(1) = a(2) = a(3) = 1, a(4) = 5, a(5) = 6, a(6) = 4; a(n) = a(n-a(n-2)) + a(n-a(n-4)) for n > 6.

Original entry on oeis.org

1, 1, 1, 5, 6, 4, 5, 6, 6, 9, 10, 5, 6, 12, 12, 15, 16, 5, 6, 18, 18, 21, 22, 5, 6, 24, 24, 27, 28, 5, 6, 30, 30, 33, 34, 5, 6, 36, 36, 39, 40, 5, 6, 42, 42, 45, 46, 5, 6, 48, 48, 51, 52, 5, 6, 54, 54, 57, 58, 5, 6, 60, 60, 63, 64, 5, 6, 66, 66, 69, 70, 5, 6, 72, 72, 75, 76, 5, 6, 78, 78, 81, 82, 5, 6

Offset: 1

Altug Alkan, May 13 2018

A quasi-periodic solution to the recurrence a(n) = a(n-a(n-2)) + a(n-a(n-4)). Although A087777 and A240809 are highly chaotic, this sequence is completely predictable thanks to its initial conditions.

Index entries for linear recurrences with constant coefficients, signature (2, -3, 4, -5, 6, -5, 4, -3, 2, -1).

Cf. A087777, A240809, A244477, A296518, A296786.

q=vector(85); q[1]=1;q[2]=1;q[3]=1;q[4]=5;q[5]=6;q[6]=4; for(n=7, #q, q[n] = q[n-q[n-2]]+q[n-q[n-4]]); q

a(6*k) = 5, a(6*k+1) = 6, a(6*k+2) = a(6*k+3) = 6*k, a(6*k+4) = 6*k+3, a(6*k+5) = 6*k+4 for k > 1.
Conjectures from Colin Barker, May 14 2018: (Start)
G.f.: x*(1 - x + 2*x^2 + 2*x^3 + 2*x^5 - x^6 + 4*x^7 - 3*x^8 + x^9 - x^10 - 2*x^11 + 2*x^12 - x^13 + x^14 + x^15 - x^16) / ((1 - x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-1) - 3*a(n-2) + 4*a(n-3) - 5*a(n-4) + 6*a(n-5) - 5*a(n-6) + 4*a(n-7) - 3*a(n-8) + 2*a(n-9) - a(n-10) for n>17.
(End)

cp's OEIS Frontend

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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A309554 a(1) = a(6) = 1, a(2) = a(3) = a(8) = 2, a(4) = a(7) = 7, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-3)) for n > 8.

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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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A296690 a(1) = a(2) = a(3) = 2, a(4) = 4, a(5) = a(6) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 6.

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A296786 a(1) = a(2) = a(5) = 2, a(3) = 1, a(4) = 3, a(6) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 6.

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A304490 a(1) = a(2) = a(3) = 1, a(4) = 5, a(5) = 6, a(6) = 4; a(n) = a(n-a(n-2)) + a(n-a(n-4)) for n > 6.

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