A309554 -id:A309554 - 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.

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.

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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