A005185 -id:A005185 - cp's OEIS frontend

A284019 The "Hofstadter chaotic heart" sequence: a(n) = A004001(n) - A005185(n).

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

Offset: 1

Altug Alkan, Mar 18 2017

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

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

Altug Alkan, Table of n, a(n) for n = 1..100000
Altug Alkan, Alternative Scatterplot of A284019
Altug Alkan, Nathan Fox, and Orhan Ozgur Aybar, On Hofstadter Heart Sequences, Complexity, 2017.
Index entries for Hofstadter-type sequences

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

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

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 *)

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])

(define (A284019 n) (- (A004001 n) (A005185 n))) ;; Needs also Scheme-code included in those two entries. - Antti Karttunen, Mar 22 2017

Graphically descriptive name added by Antti Karttunen with permission from D. R. Hofstadter, Mar 29 2017

A286560 Compound filter (summands of A004001 & summands of A005185): a(n) = P(A286541(n), A286559(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 2, 5, 41, 71, 71, 198, 313, 484, 922, 1153, 1201, 2105, 1565, 2588, 4046, 5001, 7443, 7443, 8851, 10671, 19589, 16570, 16935, 22254, 25313, 25313, 25313, 42891, 28793, 32768, 52795, 65504, 59178, 73355, 89033, 88632, 107660, 129045, 129045, 153471, 167646, 167646, 182446, 182446, 336130, 197244, 233297, 330472, 307358, 270167, 355325, 378466, 332156

Offset: 1

Antti Karttunen, May 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A004001, A005185, A284019, A286541, A286559, A286569.

(define (A286560 n) (if (<= n 2) 0 (* (/ 1 2) (+ (expt (+ (A286541 n) (A286559 n)) 2) (- (A286541 n)) (- (* 3 (A286559 n))) 2))))

a(1) = a(2) = 0, for n > 2, a(n) = (1/2)*(2 + ((A286541(n)+A286559(n))^2) - A286541(n) - 3*A286559(n)).

A283025 Remainder when sum of first n terms of A005185 is divided by n.

Original entry on oeis.org

0, 0, 1, 3, 0, 2, 5, 0, 3, 6, 9, 2, 6, 10, 1, 5, 10, 16, 3, 9, 15, 21, 4, 13, 20, 1, 9, 17, 25, 3, 14, 22, 30, 7, 18, 27, 0, 11, 21, 32, 3, 14, 26, 38, 5, 16, 27, 46, 8, 19, 35, 49, 8, 23, 38, 51, 11, 25, 41, 57, 12, 27, 50, 2, 15, 35, 52, 67, 19, 40, 58, 5, 25, 44, 64, 7, 28, 47, 67, 9, 31, 52, 73, 13, 34, 56, 80, 16, 38, 62, 86, 18

Offset: 1

Altug Alkan, Feb 27 2017

Numbers n such that a(n) = 0 are 1, 2, 5, 8, 37, 99, 1580, 42029, ...

Sequence is a mixture of regularity and irregularity. - Douglas Hofstadter, Mar 03 2017

			a(4) = 3 since Sum_{k=1..4} A005185(k) = 1 + 1 + 2 + 3 = 7 and remainder when 7 is divided by 4 is 3.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative Graph of A283025
Altug Alkan, Illustration Of Residue Classes Modulo 4

Cf. A005185, A076268, A282891, A282894.

A005185:= proc(n) option remember; procname(n-procname(n-1)) +procname(n-procname(n-2)) end proc:
A005185(1):= 1: A005185(2):= 1:
L:= ListTools[PartialSums](map(A005185, [$1..1000])):
seq(L[i] mod i, i=1..1000); # Robert Israel, Feb 28 2017

h[1]=h[2]=1; h[n_]:=h[n]= h[n-h[n-1]] + h[n-h[n-2]]; Mod[ Accumulate[h /@ Range[100]], Range[100]] (* Giovanni Resta, Feb 27 2017 *)

a=vector(1000); a[1]=a[2]=1; for(n=3, #a, a[n]=a[n-a[n-1]]+a[n-a[n-2]]); vector(#a, n, sum(k=1, n, a[k]) % n)

first(n)=my(v=vector(n), s); v[1]=v[2]=1; for(k=3, n, v[k]=v[k-v[k-1]]+v[k-v[k-2]]); for(k=1, n, s+=v[k]; v[k]=s%k); v \\ after Charles R Greathouse IV at A282891

a(n) = (Sum_{k=1..n} A005185(k)) mod n.

a(n) = A076268(n) mod n.

A283467 a(n) = A005185(n+1-A005185(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 3, 4, 5, 4, 5, 5, 5, 6, 6, 6, 6, 8, 6, 8, 8, 8, 8, 8, 10, 8, 9, 10, 10, 10, 11, 11, 10, 11, 11, 11, 12, 12, 12, 12, 12, 16, 10, 14, 14, 12, 14, 16, 14, 14, 16, 14, 16, 16, 16, 16, 20, 16, 17, 21, 16, 17, 19, 20, 20, 21, 21, 20, 19, 19, 22, 19, 21, 21, 22, 22, 22, 23, 21, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 32, 17, 32

Offset: 1

Antti Karttunen, Mar 22 2017

nonn

For n >= 2, a(n) gives the left hand summand for the term q(n+1) of Hofstadter Q-sequence (A005185): q(1) = q(2) = 1; q(n) = q(n-q(n-1)) + q(n-q(n-2)) for n > 2.

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for Hofstadter-type sequences

Cf. A005185, A280706 (partial sums), A284173.

a[1] = a[2] = 1; a[n_] := a[n] = a[n - a[n - 1]] + a[n - a[n - 2]]; Table[a[n + 1 - a[n]], {n, 97}] (* Michael De Vlieger, Mar 22 2017 *)

q(n) = if(n<3, 1, q(n - q(n - 1)) + q(n - q(n - 2)));
a(n) = q(n + 1 - q(n)); \\ Indranil Ghosh, Mar 22 2017

(define (A283467 n) (A005185 (- (+ n 1) (A005185 n)))) ;; Code for A005185 given under that entry.

a(n) = A005185(n + 1 - A005185(n)).

A286569 Restricted growth sequence transform of "Hofstadter chaotic heart", A284019 (= A004001(n) - A005185(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 2, 2, 1, 1, 1, 3, 5, 4, 3, 3, 2, 1, 1, 1, 6, 2, 1, 4, 4, 3, 3, 2, 3, 3, 3, 3, 3, 5, 5, 7, 7, 8, 9, 9, 2, 5, 9, 1, 3, 7, 2, 3, 1, 1, 1, 1, 10, 2, 5, 6, 1, 7, 4, 4, 3, 3, 1, 5, 5, 7, 3, 9, 9, 5, 5, 9, 9, 5, 9, 7, 5, 7, 11, 7, 9, 11, 11, 12, 12, 13, 14, 9, 5, 3, 15, 7, 9, 16, 4, 12, 11, 5, 1, 16, 3, 3, 17, 1, 6, 18

Offset: 1

Antti Karttunen, May 18 2017

nonn

			We start by setting a(1) = 1 for A284019(1) = 0. Then after, whenever A284019(k) is equal to some A284019(m) with m < k, we set a(k) = a(m). Otherwise (when the value is a new one, not encountered before), we allot for a(k) the least natural number not present among a(1) .. a(k-1).
For n=2, as A284019(2) = 0, which was already present at A284019(1), we set a(2) = a(1) = 1.
For n=3, as A284019(3) = 0, which was already present at n=1, we set a(3) = a(1) = 1.
For n=4, as A284019(4) = -1, which is a new value not encountered before, we set a(4) = 1 + max(a(1),a(2),a(3)) = 2.
For n=5, as A284019(5) = 0, which was already present at n=1, we set a(5) = a(1) = 1.
For n=7, as A284019(7) = -1, which was already present at n=4, we set a(7) = a(4) = 2.
For n=11, as A284019(11) = 1, which is a new value not encountered before (sign matters here), we set a(11) = 1 + max(a(1),..,a(10)) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..47000

Cf. A004001, A005185, A284019, A286560.

A064550 a(1) = 2, a(n)=a(n-1)+2*Q(n)-n, n > 1 where Q = A005185.

Original entry on oeis.org

1, 2, 4, 7, 9, 12, 16, 19, 23, 26, 28, 33, 37, 40, 46, 49, 53, 58, 62, 67, 71, 74, 76, 85, 89, 92, 98, 103, 107, 110, 120, 123, 125, 132, 140, 143, 147, 154, 158, 163, 169, 174, 180, 185, 189, 192, 194, 211, 211, 212, 222, 227, 227, 234, 240, 241

Offset: 0

Roger L. Bagula, Oct 08 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Roger L. Bagula, A Simulation of a Prime Type of Sequence: The Hofstadter Integers [broken link]

Cf. A064551, A064552, A005185.

function a064550(maxarg: integer); var n,r,rm,q: integer; qar: array; begin qar := alloc(array,maxarg + 1); qar[0] := 1; for n := 1 to maxarg do if n < 2 then q := 1; else q := qar[n - qar[n - 1]] + qar[n - qar[n - 2]]; end; qar[n] := q; if n = 1 then r := 2; else r := rm + round(2*(q - n/2)); end; rm := r; write(r," "); end; end; a064550(65);

a064550 n = a064550_list !! n
a064550_list = 1 : 2 : zipWith3 (\a q n -> a + 2 * q - n)
    (tail a064550_list) (drop 2 a005185_list) [2..]
-- Reinhard Zumkeller, May 13 2012

A064550 := proc(n) option remember; if n=0 then 1 else A064550(n-1)+2*A005185(n-1)(n) - n; fi; end;

q[0] = q[1] = 1;
q[n_] := q[n - q[n - 1]] + q[n - q[n - 2]];
a[1] = 2;
a[n_] := a[n] = a[n - 1] + 2*(q[n] - n/2);
Table[ a[n], {n, 1, 70} ]

More terms from Vladeta Jovovic, Klaus Brockhaus and Matthew Conroy, Oct 09 2001

A076268 Sum(k=1,n, A005185(k)).

Original entry on oeis.org

1, 2, 4, 7, 10, 14, 19, 24, 30, 36, 42, 50, 58, 66, 76, 85, 95, 106, 117, 129, 141, 153, 165, 181, 195, 209, 225, 241, 257, 273, 293, 310, 327, 347, 368, 387, 407, 429, 450, 472, 495, 518, 542, 566, 590, 614, 638, 670, 694, 719, 749, 777, 803, 833, 863, 891

Offset: 1

Benoit Cloitre, Nov 05 2002

nonn

Partial sums of Hofstadter Q-sequence. The subsequence of primes in the partial sum begins: 2, 7, 19, 181, 241, 257, 293, 347, 719, 863. The subsequence of square in the partial sum begins: 1, 4, 36, 225. [From Jonathan Vos Post, Apr 09 2010]

Cf. A005185.

A076701 Values of n not reached in the Hofstadter sequence (A005185).

Original entry on oeis.org

7, 13, 15, 18, 27, 29, 34, 36, 49, 51, 59, 67, 70, 74, 81, 89, 95, 97, 98, 99, 102, 103, 117, 126, 127, 131, 134, 141, 142, 145, 150, 158, 163, 166, 181, 183, 189, 191, 195, 197, 198, 199, 205, 207, 209, 213, 224, 225, 232, 247, 259, 265, 267, 270, 274, 281

Offset: 1

Benoit Cloitre, Oct 26 2002

nonn

For any n, there is no solution to a(n) = A005185(x). Does lim_{n -> infinity} a(n)/n exist?

Plotting a(n)/n for the first 6 billion terms, it appears to be converging to ~7.39. - Benjamin Chaffin, Sep 17 2019

Benjamin Chaffin, Table of n, a(n) for n = 1..13805
Benjamin Chaffin, Log scale plot of a(n)/n from 1 .. 6*10^9
Benjamin Chaffin, Log scale plot of a(n)/n from 1000000 .. 6*10^9

Cf. A005185.

a[1] = a[2] = 1; a[n_] := a[n] = a[n - a[n - 1]] + a[n - a[n - 2]]; t = Table[a[n], {n, 1000}]; Take[ Complement[ Range@ 502, Union@ t], 56]

A087740 a(n) = 1 + abs(A004001(A005185(n)) - A005185(A004001(n))).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 2, 2, 2, 4, 1, 2, 2, 2, 2, 1, 2, 1, 2, 3, 3, 1, 1, 1, 3, 3, 1, 3, 2, 1, 1, 2, 1, 5, 5, 5, 5, 2, 2, 2, 2, 7, 1, 2, 2, 3, 3, 3, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 3, 2, 2, 6, 7, 7, 4, 3, 4, 4, 2, 2, 2, 4, 4, 3, 7, 3, 3, 2, 6, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 1, 3, 1, 3, 2, 9, 5, 9, 10

Offset: 1

Roger L. Bagula, Oct 01 2003

A "commutator" between the Hofstadter A005185 sequence and the Conway-Hofstadter A004001 sequence.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A004001, A005185, A284019 (compare the scatter plots).

Cf. also A286560.

Conway[n_Integer?Positive] := Conway[n] =Conway[Conway[n-1]] + Conway[n - Conway[n-1]] Conway[1] = Conway[2] = 1 Hofstadter[n_Integer?Positive] := Hofstadter[n] = Hofstadter[n - Hofstadter[n-1]] + Hofstadter[n - Hofstadter[n-2]] Hofstadter[1] = Hofstadter[2] = 1 digits=200 a=Table[1+Abs[Conway[Hofstadter[n]]-Hofstadter[Conway[n]]], {n, 1, digits}]

(define (A087740 n) (+ 1 (abs (- (A004001 (A005185 n)) (A005185 (A004001 n)))))) ;; Scheme-code for A004001 and A005185 given under those entries.

Data section extended to 120 terms by Antti Karttunen, May 22 2017

A226244 Successive record-setters (maxima) in Hofstadter's Q-sequence (A005185).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 16, 20, 21, 22, 23, 24, 32, 40, 42, 43, 44, 46, 47, 48, 64, 66, 68, 72, 78, 80, 82, 83, 85, 88, 90, 92, 94, 96, 128, 130, 138, 149, 151, 152, 159, 162, 165, 168, 169, 170

Offset: 1

Jeffrey Shallit, Jun 01 2013

nonn

a(n) = A005185(A226245(n)).

			a(8) = 10 because the first few terms of Hofstadter's Q-sequence are 1,1,2,3,3,4,5,5,6,6,6,8,8,8,10,9 and 10 is the 8th record value.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A005185

a226244 n = a226244_list !! (n-1)
(a226244_list, a226245_list) = unzip $ (1,1) : f 1 1 a005185_list where
   f i v (q:qs) | q > v = (q,i) : f (i + 1) q qs
                | otherwise = f (i + 1) v qs
-- Reinhard Zumkeller, Jun 02 2013

cp's OEIS Frontend

A284019 The "Hofstadter chaotic heart" sequence: a(n) = A004001(n) - A005185(n).

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A286560 Compound filter (summands of A004001 & summands of A005185): a(n) = P(A286541(n), A286559(n)), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = a(2) = 0.

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A283025 Remainder when sum of first n terms of A005185 is divided by n.

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A283467 a(n) = A005185(n+1-A005185(n)).

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A286569 Restricted growth sequence transform of "Hofstadter chaotic heart", A284019 (= A004001(n) - A005185(n)).

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A064550 a(1) = 2, a(n)=a(n-1)+2*Q(n)-n, n > 1 where Q = A005185.

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A076268 Sum(k=1,n, A005185(k)).

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A076701 Values of n not reached in the Hofstadter sequence (A005185).

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