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A284019 The "Hofstadter chaotic heart" sequence: a(n) = A004001(n) - A005185(n). 7
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,15
COMMENTS
See also scatterplot in Links section.
From Nathan Fox, Mar 30 2017: (Start)
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.
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]
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
LINKS
Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, 2017.
EXAMPLE
a(4) = -1 since a(4) = A004001(4) - A005185(4) = 2 - 3 = -1.
MAPLE
A005185:= proc(n) option remember; procname(n-procname(n-1)) +procname(n-procname(n-2)) end proc:
A005185(1):= 1: A005185(2):= 1:
A004001:= proc(n) option remember; procname(procname(n-1)) +procname(n-procname(n-1)) end proc:
A004001(1):= 1: A004001(2):= 1:
A284019:= map(A004001 - A005185, [$1..1000]):
seq(A284019[i], i=1..1000); # Altug Alkan, Mar 31 2017
MATHEMATICA
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 *)
PROG
(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])
(Scheme) (define (A284019 n) (- (A004001 n) (A005185 n))) ;; Needs also Scheme-code included in those two entries. - Antti Karttunen, Mar 22 2017
CROSSREFS
Sequence in context: A026836 A089052 A284606 * A286135 A142475 A051556
KEYWORD
sign,look
AUTHOR
Altug Alkan, Mar 18 2017
EXTENSIONS
Graphically descriptive name added by Antti Karttunen with permission from D. R. Hofstadter, Mar 29 2017
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)