cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Altug Alkan, Mar 18 2017

Keywords

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

Examples

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

Links

Crossrefs

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

Programs

  • 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 *)
  • 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

Extensions

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

A286541 Compound filter (the left & right summand of Hofstadter-Conway $10000 sequence): a(n) = P(A004001(A004001(n-1)), A004001(n-A004001(n-1))), 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, 1, 2, 5, 5, 5, 8, 13, 19, 19, 25, 25, 25, 25, 32, 41, 51, 62, 62, 73, 86, 86, 99, 99, 99, 113, 113, 113, 113, 113, 128, 145, 163, 182, 202, 202, 222, 244, 267, 267, 290, 315, 315, 340, 340, 340, 366, 394, 394, 422, 422, 422, 451, 451, 451, 451, 481, 481, 481, 481, 481, 481, 512, 545, 579, 614, 650, 687, 687, 724, 763, 803, 844, 844, 885, 928, 972, 972
Offset: 1

Views

Author

Antti Karttunen, May 18 2017

Keywords

Links

Crossrefs

Cf. A004001, A286559, A286560.
Cf. A283468, A283469, A283470, A283472 and also A283471, A283473.

Programs

Formula

a(1) = a(2) = 0, for n > 2, a(n) = (1/2)*(2 + ((A004001(A004001(n-1))+A004001(n-A004001(n-1)))^2) - A004001(A004001(n-1)) - 3*A004001(n-A004001(n-1))).

A286559 Compound filter (the left & right summand of Hofstadter Q-sequence): a(n) = P(Q(n-Q(n-1)), Q(n-Q(n-2))), 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, 2, 5, 8, 8, 13, 13, 13, 25, 24, 25, 41, 32, 41, 50, 50, 61, 61, 61, 61, 113, 84, 86, 113, 113, 113, 113, 181, 128, 129, 181, 200, 163, 182, 221, 200, 221, 242, 242, 265, 265, 265, 265, 265, 481, 263, 290, 420, 363, 314, 422, 420, 365, 481, 420, 481, 481, 481, 481, 761, 512, 452, 687, 577, 513, 722, 761, 650, 687, 762, 723, 760, 722, 842, 760, 801
Offset: 1

Views

Author

Antti Karttunen, May 18 2017

Keywords

Links

Crossrefs

Cf. A005185, A283467, A286541, A286560.

Programs

Formula

a(1) = a(2) = 0, for n > 2, a(n) = (1/2)*(2 + ((A005185(n-A005185(n-1))+A005185(n-A005185(n-2)))^2) - A005185(n-A005185(n-1)) - 3*A005185(n-A005185(n-2))).

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

Views

Author

Antti Karttunen, May 18 2017

Keywords

Examples

			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.

Links

Crossrefs

Cf. A004001, A005185, A284019, A286560.

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

Views

Author

Roger L. Bagula, Oct 01 2003

Keywords

Comments

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

Links

Crossrefs

Cf. A004001, A005185, A284019 (compare the scatter plots).
Cf. also A286560.

Programs

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

Extensions

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

A302780 Restricted growth sequence transform of 4-tuple [H(H(n-1)), H(n-H(n-1)), Q(n-Q(n-1)), Q(n-Q(n-2))] where H = A004001 and Q = A005185.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 25, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 47, 50, 50, 50, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 79, 80, 80
Offset: 1

Views

Author

Antti Karttunen, Apr 27 2018

Keywords

Comments

Restricted growth sequence transform of A286560: a filter sequence which includes both the summands of A004001 and the summands of A005185.
For all i, j: a(i) = a(j) => b(i) = b(j), where b is a sequence like A087740, A284019, A286569 or A302779.
For n > 1000 the duplicates get rare. In range [1000, 65536] there are only three cases: a(1353) = a(1354) = 1319, a(39361) = a(39362) = 39326, and a(46695) = a(46696) = 46659.

Links

Crossrefs

Cf. A004001, A005185, A286560.
Cf. also A087740, A284019, A286541, A286569, A302779.

Programs

  • PARI
    up_to = 65537;
first_n_of_A004001(n) = { my(v=vector(n)); v[1]=v[2]=1; for(k=3, n, v[k]=v[v[k-1]]+v[k-v[k-1]]); (v); }; \\ Charles R Greathouse IV, Feb 26 2017
v004001 = first_n_of_A004001(up_to);
A004001(n) = v004001[n];
first_n_of_A005185(n) = { my(v=vector(n)); v[1]=v[2]=1; for(k=3, n, v[k]=v[k-v[k-1]]+v[k-v[k-2]]); (v); }; \\
v005185 = first_n_of_A005185(up_to);
A005185(n) = v005185[n];
Aux302780(n) = if(n<3,0,[A004001(A004001(n-1)), A004001(n-A004001(n-1)), A005185(n-A005185(n-1)), A005185(n-A005185(n-2))]);
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux302780(n))),"b302780.txt");
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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