A272610 -id:A272610 - cp's OEIS frontend

A272611 a(1)=1; thereafter a(n) = a(n-a(n-1)) + A272612(n-1).

1, 2, 3, 3, 4, 5, 5, 7, 5, 8, 8, 7, 10, 8, 10, 12, 9, 11, 15, 9, 14, 16, 11, 19, 13, 18, 13, 19, 19, 18, 19, 20, 20, 18, 19, 20, 20, 23, 27, 20, 26, 28, 20, 35, 20, 31, 27, 26, 26, 37, 21, 33, 24, 35, 23, 37, 24, 35, 38, 33, 34, 39, 39, 29, 39, 32, 38, 39, 37

Offset: 1

Nathan Fox, May 03 2016

nonn

Much like the Hofstadter Q-sequence A005185, it is not known if this sequence is defined for all positive n.
Empirically, this sequence appears to grow approximately like n/2 with a lot of noise.
a(n) exists for n<=10^7.

Nathan Fox, Table of n, a(n) for n = 1..10000

Cf. A005185, A272610, A272612, A272613.

#Code for A272611, A272612, and A272613
A272611:=proc(n) option remember:
    if n = 1 then
        return 1:
    else
        return A272611(n-A272611(n-1))+A272612(n-1):
    fi:
end:
A272612:=proc(n) option remember:
    if n = 0 then
        return 1:
    elif n = 1 then
        return 1:
    else
        return A272612(n-A272611(n))+A272612(n-A272611(n-1)):
    fi:
end:
A272613:=proc(n) option remember:
    if n = 0 then
        return 1:
    else
        return A272613(n-A272611(n))+A272613(n-A272612(n)):
    fi:
end:

A272612 a(0)=1, a(1)=1; thereafter a(n) = a(n-A272611(n)) + a(n-A272611(n-1)).

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 4, 3, 4, 5, 4, 5, 5, 5, 7, 5, 6, 8, 5, 7, 9, 6, 9, 8, 8, 8, 9, 11, 10, 9, 10, 10, 10, 10, 11, 11, 12, 15, 10, 12, 16, 10, 16, 12, 13, 15, 12, 15, 18, 13, 14, 15, 17, 14, 17, 15, 17, 19, 17, 15, 20, 20, 16, 19, 19, 19, 21, 19, 19, 20, 20, 20, 20

Offset: 0

Nathan Fox, May 03 2016

nonn

Much like the Hofstadter Q-sequence A005185, it is not known if this sequence is defined for all positive n.
Empirically, this sequence appears to grow approximately like n/4 with a lot of noise.
a(n) exists for n<=10^7.

Nathan Fox, Table of n, a(n) for n = 0..10000

Cf. A005185, A272610, A272611, A272613.

Maple
```
See A272611 for Maple code
```

A272613 a(0)=1; thereafter a(n) = a(n-A272611(n)) + a(n-A272612(n)).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 5, 6, 8, 6, 8, 9, 9, 13, 10, 12, 14, 11, 17, 17, 14, 17, 22, 16, 23, 17, 24, 22, 23, 23, 23, 26, 31, 28, 28, 35, 37, 32, 32, 39, 33, 39, 46, 34, 49, 33, 45, 48, 45, 46, 60, 54, 58, 56, 58, 50, 70, 54, 53, 73, 57, 61, 61, 77, 50, 73, 56, 68, 72

Offset: 0

Nathan Fox, May 03 2016

nonn

This sequence exists as long as A272611 and A272612 exist.

Empirically, this sequence appears to grow approximately like n^a for some a between 1 and 2.

Nathan Fox, Table of n, a(n) for n = 0..10000

Cf. A005185, A272610, A272611, A272612.

Maple
```
See A272611 for Maple code
```

A283903 Relative of Hofstadter Q-sequence.

Original entry on oeis.org

11, 20, 6, 6, 6, 20, 11, 20, 6, 6, 6, 20, 11, 6, 12, 12, 12, 20, 11, 6, 18, 18, 12, 40, 11, 12, 24, 18, 12, 40, 11, 18, 30, 18, 18, 40, 11, 24, 30, 18, 18, 80, 11, 30, 30, 24, 24, 80, 11, 30, 36, 30, 24, 80, 11, 30, 48, 24, 24, 80, 11, 36, 54, 24, 24, 160, 11, 48, 42, 36, 30, 120, 11, 54, 42

Offset: 1

Nathan Fox, Mar 19 2017

This sequence is defined by a(n) = 0 for n <= 0; a(1) = 11, a(2) = 20, a(3) = 6, a(4) = 6, a(5) = 6, a(6) = 20, a(7) = 11, a(8) = 20, a(9) = 6, a(10) = 6, a(11) = 6, a(12) = 20, a(13) = 11, a(14) = 6, a(15) = 12, a(16) = 12, a(17) = 12, a(18) = 20; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).
Similar to Hofstadter's Q-sequence A005185 but with different starting values.
For as long as it exists, this sequence has a similar structure to A272160. That sequence consists of five interleaved sequences: four chaotic sequences and a sequence of all 4's. This sequence consists of six interleaved sequences: five chaotic sequences and a sequence of all 11's.
If the 20's in the initial condition are each replaced by larger numbers, the general structure of this sequence does not change.
This sequence has exactly 2179 terms, since a(2179)=0 and computing a(2180) would refer to itself.

Nathan Fox, Table of n, a(n) for n = 1..2179

Cf. A005185, A272610, A284053, A284054.

A283903:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 11: elif n = 2 then 20: elif n = 3 then 6: elif n = 4 then 6: elif n = 5 then 6: elif n = 6 then 20: elif n = 7 then 11: elif n = 8 then 20: elif n = 9 then 6: elif n = 10 then 6: elif n = 11 then 6: elif n = 12 then 20: elif n = 13 then 11: elif n = 14 then 6: elif n = 15 then 12: elif n = 16 then 12: elif n = 17 then 12: elif n = 18 then 20: else A283903(n-A283903(n-1)) + A283903(n-A283903(n-2)): fi: end:

A284053 Relative of Hofstadter Q-sequence.

Original entry on oeis.org

9, 20, 5, 5, 20, 9, 20, 5, 5, 20, 9, 5, 10, 10, 20, 9, 5, 15, 15, 40, 9, 10, 20, 15, 40, 9, 15, 25, 15, 40, 9, 20, 25, 15, 60, 9, 25, 25, 20, 60, 9, 25, 30, 25, 60, 9, 25, 40, 20, 60, 9, 30, 45, 20, 80, 9, 40, 35, 30, 80, 9, 45, 35, 30, 100, 9, 35, 55, 25, 80, 9, 35, 55, 35, 100

Offset: 1

Nathan Fox, Mar 19 2017

nonn

This sequence is defined by a(n) = 0 for n <= 0; a(1) = 9, a(2) = 20, a(3) = 5, a(4) = 5, a(5) = 20, a(6) = 9, a(7) = 20, a(8) = 5, a(9) = 5, a(10) = 20, a(11) = 9, a(12) = 5, a(13) = 10, a(14) = 10, a(15) = 20; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).
Similar to Hofstadter's Q-sequence A005185 but with different starting values.
Much like the Hofstadter Q-sequence A005185, it is not known if this sequence is defined for all positive n.
This sequence has a similar structure to A272610. That sequence consists of five interleaved sequences: four chaotic sequences and a sequence of all 4's. This sequence also consists of five interleaved sequences: four chaotic sequences and a sequence of all 9's.
If the 20's in the initial condition are each replaced by larger numbers, the general structure of this sequence does not change.

Nathan Fox, Table of n, a(n) for n = 1..10000

Cf. A005185, A272610, A283903, A284054.

A284053:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 9: elif n = 2 then 20: elif n = 3 then 5: elif n = 4 then 5: elif n = 5 then 20: elif n = 6 then 9: elif n = 7 then 20: elif n = 8 then 5: elif n = 9 then 5: elif n = 10 then 20: elif n = 11 then 9: elif n = 12 then 5: elif n = 13 then 10: elif n = 14 then 10: elif n = 15 then 20: else A284053(n-A284053(n-1)) + A284053(n-A284053(n-2)): fi: end:

from functools import cache
@cache
def a(n):
    if n <= 0: return 0
    if n < 16:
        return [9, 20, 5, 5, 20, 9, 20, 5, 5, 20, 9, 5, 10, 10, 20][n-1]
    return a(n - a(n-1)) + a(n - a(n-2))
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Sep 20 2021

A284054 Relative of Hofstadter Q-sequence.

Original entry on oeis.org

7, 26, 8, 8, 8, 8, 8, 26, 7, 8, 16, 16, 16, 16, 16, 26, 7, 16, 24, 8, 16, 24, 8, 26, 7, 24, 16, 24, 24, 16, 24, 52, 7, 16, 48, 8, 24, 32, 24, 52, 7, 48, 8, 8, 48, 32, 16, 78, 7, 8, 16, 16, 48, 40, 24, 78, 7, 16, 24, 16, 72, 32, 24, 104, 7, 24, 24, 16, 96, 40, 24, 130, 7, 24, 32

Offset: 1

Nathan Fox, Mar 19 2017

nonn

This sequence is defined by a(n) = 0 for n <= 0; a(1) = 7, a(2) = 26, a(3) = 8, a(4) = 8, a(5) = 8, a(6) = 8, a(7) = 8, a(8) = 26, a(9) = 7, a(10) = 8, a(11) = 16, a(12) = 16, a(13) = 16, a(14) = 16, a(15) = 16, a(16) = 26; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).
Similar to Hofstadter's Q-sequence A005185 but with different starting values.
Much like the Hofstadter Q-sequence A005185, it is not known if this sequence is defined for all positive n.
This sequence has a similar structure to A272160. That sequence consists of five interleaved sequences: four chaotic sequences and a sequence of all 4's. This sequence appears to consist eventually of eight interleaved sequences: four chaotic sequences, a sequence of all 7's, a sequence of mostly 32's and an few 40's, a sequence of all 24's, and a rapidly growing sequence with successive terms satisfying either the recurrence A(k) = A(k-3) + A(k-4) or the recurrence A(k) = A(k-3) + A(k-5).
If the 26's in the initial condition are each replaced by larger numbers, the general structure of this sequence does not change.

Nathan Fox, Table of n, a(n) for n = 1..10000

Cf. A005185, A272610, A283903, A284053.

A284054:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 7: elif n = 2 then 26: elif n = 3 then 8: elif n = 4 then 8: elif n = 5 then 8: elif n = 6 then 8: elif n = 7 then 8: elif n = 8 then 26: elif n = 9 then 7: elif n = 10 then 8: elif n = 11 then 16: elif n = 12 then 16: elif n = 13 then 16: elif n = 14 then 16: elif n = 15 then 16: elif n = 16 then 26: else A284054(n-A284054(n-1)) + A284054(n-A284054(n-2)): fi: end:

from functools import lru_cache
@lru_cache(maxsize=None)
def a(n):
    if n <= 0: return 0
    if n < 17:
        return [7, 26, 8, 8, 8, 8, 8, 26, 7, 8, 16, 16, 16, 16, 16, 26][n-1]
    return a(n - a(n-1)) + a(n - a(n-2))
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Jul 26 2021

A274055 Relative of Hofstadter Q-sequence: a(n) = n for 1 <= n <= 42; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 42.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 3, 43, 44, 5, 45, 6, 7, 46, 48, 10, 8, 48, 52, 12, 49, 14, 54, 11, 53, 57, 16, 13, 17, 15, 56, 20, 20

Offset: 1

Nathan Fox, Nov 13 2016

nonn

In calculating terms of this sequence, use the convention that a(n)=0 for n<=0.

This sequence eventually settles into a pattern resembling A272610.

Nathan Fox, Table of n, a(n) for n = 1..40000
N. Fox, Hofstadter-like Sequences over Nonstandard Integers", Talk given at the Rutgers Experimental Mathematics Seminar, November 10 2016.

Cf. A005185, A272610, A272611, A272612, A272613, A278056, A278057, A278058, A278059, A278060, A278061, A278062, A278063, A278064, A278065.

If the index is between 77 and 89 (inclusive), then a(5n) = 3, a(5n+1) = 5, a(5n+2) = 88n-1188, a(5n+3) = 5, a(5n+4) = 88.
If the index is between 95 and 397 (inclusive), then a(5n) = 396n-6820, a(5n+1) = 3, a(5n+2) = 396, a(5n+3) = 3, a(5n+4) = 5.
If the index is between 403 and 24860 (inclusive), then a(5n) = 24860, a(5n+1) = 3, a(5n+2) = 5, a(5n+3) = 24860n-1939476, a(5n+4) = 5.
If the index is at least 24863, then a(5n) = 24860*A272613(n-4972), a(5n+1) = 4, a(5n+2) = 5*A272611(n-4972), a(5n+3) = 5*A272611(n-4971), a(5n+4) = 5*A272612(n-4971). This pattern lasts as long as A272611 exists (which is conjectured to be forever).

A283882 Relative of Hofstadter Q-sequence: a(n) = max(0, n+67) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 0.

Original entry on oeis.org

3, 68, 69, 5, 70, 6, 7, 71, 73, 10, 8, 73, 77, 12, 74, 14, 79, 11, 78, 82, 16, 13, 17, 15, 81, 20, 20, 142, 73, 24, 32, 138, 3, 32, 207, 5, 138, 3, 5, 345, 5, 138, 3, 5, 483, 5, 138, 3, 5, 621, 5, 138, 3, 5, 759, 5, 138, 3, 5, 897, 5, 138, 3, 5, 1035, 5, 138, 3, 5, 1173, 5, 138, 5, 8, 1311

Offset: 1

Nathan Fox, Mar 19 2017

nonn

Sequences like this are more naturally considered with the first nonzero term in position 1. But this sequence would then match A000027 for its first 67 terms.

Nathan Fox, Table of n, a(n) for n = 1..10000

Cf. A005185, A272610, A272611, A272612, A272613, A274058, A283883.

A283882:=proc(n) option remember: if n <= 0 then max(0, n+67): else A283882(n-A283882(n-1)) + A283882(n-A283882(n-2)): fi: end:

If the index is between 35 and 72 (inclusive), then a(5n) = 138n-759, a(5n+1) = 5, a(5n+2) = 138, a(5n+3) = 3, a(5n+4) = 5.
If the index is between 78 and 1245 (inclusive), then a(5n) = 1311, a(5n+1) = 3, a(5n+2) = 5, a(5n+3) = 1311n-17181, a(5n+4) = 5.
If the index is between 1251 and 309192 (inclusive), then a(5n) = 5, a(5n+1) = 19047817435n-1178393232110703, a(5n+2) = 5, a(5n+3) = 19047817435, a(5n+4) = 3.
If the index is between 309336 and 19047817368 (inclusive), then a(5n) = 5, a(5n+1) = 309258n-76697295, a(5n+2) = 5, a(5n+3) = 309258, a(5n+4) = 3.
If the index is at least 19047817371, then a(5n) = 5*A272611(n-3809563474), a(5n+1) = 5*A272611(n-3809563473), a(5n+2) = 5*A272612(n-3809563473), a(5n+3) = 19047817435*A272613(n-3809563473), a(5n+4) = 4. This pattern lasts as long as A272611 exists (which is conjectured to be forever).

cp's OEIS Frontend

A272611 a(1)=1; thereafter a(n) = a(n-a(n-1)) + A272612(n-1).

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A272612 a(0)=1, a(1)=1; thereafter a(n) = a(n-A272611(n)) + a(n-A272611(n-1)).

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A272613 a(0)=1; thereafter a(n) = a(n-A272611(n)) + a(n-A272612(n)).

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A283903 Relative of Hofstadter Q-sequence.

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A284053 Relative of Hofstadter Q-sequence.

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A284054 Relative of Hofstadter Q-sequence.

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A274055 Relative of Hofstadter Q-sequence: a(n) = n for 1 <= n <= 42; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 42.

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A283882 Relative of Hofstadter Q-sequence: a(n) = max(0, n+67) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 0.

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