A284019 - OEIS

%I #75 Oct 30 2017 03:48:40

%S 0,0,0,-1,0,0,-1,-1,-1,0,1,-1,0,0,-2,-1,-1,-1,0,0,0,1,2,-2,1,1,-1,0,0,

%T 0,-4,-1,0,-2,-2,1,1,-1,1,1,1,1,1,2,2,3,3,-5,4,4,-1,2,4,0,1,3,-1,1,0,

%U 0,0,0,-8,-1,2,-4,0,3,-2,-2,1,1,0,2,2,3,1,4,4,2,2,4,4,2,4,3,2

%N The "Hofstadter chaotic heart" sequence: a(n) = A004001(n) - A005185(n).

%C See also scatterplot in Links section.

%C From _Nathan Fox_, Mar 30 2017: (Start)

%C The pattern in the graph presumably comes from the known pattern in the Conway sequence minus n/2 (A004001) combined with the "sausage" pattern of the Q-sequence (A005185). Since the Q-sequence seems to remain close to n/2, the patterns combine in this way.

%C This means that the bottoms of the hearts should be roughly at powers of 2 and the joins between them should be where the sausages thin out. (End) [Corrected by _Altug Alkan_, Apr 01 2017]

%C Note that this comment says that the indices where the bottoms of the hearts occur (the local minima) are roughly powers of 2. For example, a(8056) = -317 is a local minimum close to 2^13. - _N. J. A. Sloane_, Apr 01 2017

%H Altug Alkan, <a href="/A284019/b284019.txt">Table of n, a(n) for n = 1..100000</a>

%H Altug Alkan, <a href="/A284019/a284019_1.png">Alternative Scatterplot of A284019</a>

%H Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, <a href="https://www.hindawi.com/journals/complexity/aip/2614163/">On Hofstadter Heart Sequences</a>, Complexity, 2017.

%H <a href="/index/Ho#Hofstadter">Index entries for Hofstadter-type sequences</a>

%e a(4) = -1 since a(4) = A004001(4) - A005185(4) = 2 - 3 = -1.

%p A005185:= proc(n) option remember; procname(n-procname(n-1)) +procname(n-procname(n-2)) end proc:

%p A005185(1):= 1: A005185(2):= 1:

%p A004001:= proc(n) option remember; procname(procname(n-1)) +procname(n-procname(n-1)) end proc:

%p A004001(1):= 1: A004001(2):= 1:

%p A284019:= map(A004001 - A005185, [$1..1000]):

%p seq(A284019[i], i=1..1000); # _Altug Alkan_, Mar 31 2017

%t a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; b[1] = b[2] = 1; b[n_] := b[n] = b[n - b[n - 1]] + b[n - b[n - 2]]; Table[a@ n - b@ n, {n, 87}] (* _Michael De Vlieger_, Mar 18 2017, after _Robert G. Wilson v_ at A004001 *)

%o (PARI) q=vector(1000); h=vector(1000); q[1]=q[2]=1; for(n=3, #q, q[n]=q[n-q[n-1]]+q[n-q[n-2]]); h[1]=h[2]=1; for(n=3, #h, h[n]=h[h[n-1]]+h[n-h[n-1]]); vector(1000, n, h[n]-q[n])

%o (Scheme) (define (A284019 n) (- (A004001 n) (A005185 n))) ;; Needs also Scheme-code included in those two entries. - _Antti Karttunen_, Mar 22 2017

%Y Cf. A004001, A005185, A284606, A286560, A286569.

%K sign,look

%O 1,15

%A _Altug Alkan_, Mar 18 2017

%E Graphically descriptive name added by _Antti Karttunen_ with permission from D. R. Hofstadter, Mar 29 2017