A309567 - OEIS

%I #26 May 09 2020 00:45:00

%S 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,

%T 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,

%U 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

%N 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.

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

%H Robert Israel, <a href="/A309567/b309567.txt">Table of n, a(n) for n = 1..10000</a>

%H Altug Alkan, Nathan Fox, Orhan Ozgur Aybar, Zehra Akdeniz, <a href="https://arxiv.org/abs/2002.03396">On Some Solutions to Hofstadter's V-Recurrence</a>, arXiv:2002.03396 [math.DS], 2020.

%F For k >= 1:

%F a(5*k)   = 5*floor(sqrt(k-1))+1,

%F a(5*k+1) = 5*round(sqrt(k))-1,

%F a(5*k+2) = 5*k+2,

%F a(5*k+3) = 5,

%F a(5*k+4) = 5*k+3.

%p f:= proc(n) local k,j;

%p   j:= n mod 5;

%p   k:= (n-j)/5;

%p   if j=0 then 5*floor(sqrt(k-1))+1

%p   elif j=1 then 5*round(sqrt(k))-1

%p   elif j=2 then 5*k+2

%p   elif j=3 then 5

%p   else 5*k+3

%p   fi

%p end proc:

%p f(1):= 4:

%p map(f, [$1..100]); # _Robert Israel_, Aug 08 2019

%t 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 *)

%o (PARI) 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

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

%K nonn,easy

%O 1,1

%A _Altug Alkan_ and _Rémy Sigrist_, Aug 08 2019