A272613 -id:A272613 - cp's OEIS frontend

A272610 a(1)=5, a(2)=9, a(3)=4, a(4)=6; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

5, 9, 4, 6, 5, 5, 18, 4, 5, 10, 10, 18, 4, 10, 15, 10, 27, 4, 15, 15, 10, 36, 4, 15, 20, 15, 36, 4, 20, 25, 15, 45, 4, 25, 25, 20, 45, 4, 25, 35, 15, 54, 4, 35, 25, 20, 72, 4, 25, 40, 25, 54, 4, 40, 40, 20, 72, 4, 40, 35, 25, 81, 4, 35, 50, 25, 81, 4, 50, 40, 25, 117

Offset: 1

Nathan Fox, May 03 2016

nonn

In calculating terms of this sequence, use the convention that a(n)=0 for n<=0.
Similar to Hofstadter's Q-sequence A005185 but with different starting values.
This sequence exists as long as A272611 and A272612 exist.
No other term of this sequence changes if a(4) is replaced by a number greater than 6.
If a(2) is replaced by a number N greater than 9, then every other term of the form a(5n+2) is replaced by a(5n+2)*N/9.

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

Cf. A005185, A272611, A272612, A272613.

A272610:=proc(n) option remember:
    if n <= 0 then
        return 0:
    elif n = 1 then
        return 5:
    elif n = 2 then
        return 9:
    elif n = 3 then
        return 4:
    elif n = 4 then
        return 6:
    else
        return A272610(n-A272610(n-1))+A272610(n-A272610(n-2)):
    fi:
end:

a[n_] := a[n] = Switch[n, ?NonPositive, 0, 1, 5, 2, 9, 3, 4, 4, 6, ,
   a[n - a[n - 1]] + a[n - a[n - 2]]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jul 24 2022 *)

from functools import cache
@cache
def a(n):
    if n < 0: return 0
    if n < 5: return [0, 5, 9, 4, 6][n]
    return a(n - a(n-1)) + a(n - a(n-2))
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Sep 20 2021

a(1)=5, a(2)=9, a(3)=4, a(4)=6; thereafter a(5n)=5*A272611(n), a(5n+1)=5*A272612(n), a(5n+2)=9*A272613(n), a(5n+3)=4, a(5n+4)=5*A272611(n).

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

Original entry on oeis.org

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
```

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

A272610 a(1)=5, a(2)=9, a(3)=4, a(4)=6; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

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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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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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