A278066 -id:A278066 - cp's OEIS frontend

A278067 Relative of Hofstadter Q-sequence: a(1) = 37, a(2) = 2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 2, 37, 39, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 2, 76, 39

Offset: 1

Nathan Fox, Nov 13 2016

In calculating terms of this sequence, use the convention that a(n)=0 for n<=0.
This sequence has exactly 3850 terms, since a(3850)=0 and computing a(3851) would refer to itself.
Superficially, this sequence behaves similarly to A278066. But, that sequence is infinite, but this sequence dies.

Nathan Fox, Table of n, a(n) for n = 1..3850
Nathan Fox, Hofstadter-like Sequences over Nonstandard Integers", Talk given at the Rutgers Experimental Mathematics Seminar, November 10 2016.

Cf. A005185, A278066, A278068.

a[1] = 37; a[2] = 2; a[n_] := a[n] = If[n < 1, 0, a[n-a[n-1]] + a[n-a[n-2]]];
Array[a, 100] (* Paolo Xausa, May 30 2024 *)

A278068 Relative of Hofstadter Q-sequence: a(1) = 57, a(2) = 2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

Original entry on oeis.org

57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 2, 57, 59, 2, 116, 2, 116, 2, 116, 2, 116, 2, 116, 2, 116, 2, 116

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 is eventually, beginning with a(3128), quasilinear with quasiperiod 1402.

Nathan Fox, Table of n, a(n) for n = 1..10000
Nathan Fox, Hofstadter-like Sequences over Nonstandard Integers", Talk given at the Rutgers Experimental Mathematics Seminar, November 10 2016.
Nathan Fox, Formula for a(n)

Cf. A005185, A275163, A278066, A278067.

a[1] = 57; a[2] = 2; a[n_] := a[n] = If[n < 1, 0, a[n-a[n-1]] + a[n-a[n-2]]];
Array[a, 100] (* Paolo Xausa, May 31 2024 *)

A283884 Relative of Hofstadter Q-sequence: a(n) = max(0, n+193) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

Original entry on oeis.org

6, 194, 195, 196, 9, 197, 198, 199, 12, 200, 201, 202, 15, 203, 204, 17, 206, 18, 206, 208, 209, 22, 21, 397, 391, 9, 18, 406, 409, 202, 22, 223, 228, 206, 27, 36, 230, 396, 197, 39, 231, 237, 201, 42, 233, 240, 16, 232, 240, 220, 40

Offset: 1

Nathan Fox, Mar 19 2017

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 193 terms.

Most terms in this sequence appear in long period-5 quasilinear runs. These runs are separated by 441 other terms, and each run is approximately six times as long as the previous.

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

Cf. A005185, A267501, A274058, A278055, A278066, A283885, A283886, A283887, A283888.

A283884:=proc(n) option remember: if n <= 0 then max(0, n+193): else A283884(n-A283884(n-1)) + A283884(n-A283884(n-2)) + A283884(n-A283884(n-3)): fi: end:

If the index is between 67 and 195 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+195, a(7n+2) = 7n+197, a(7n+3) = 7, a(7n+4) = 2n+431, a(7n+5) = n+379, a(7n+6) = 191.
For nonnegative integers i, if 1<=5n+r<=(2417/5)*6^(i+1)-3382/5, then
a((2417/5)*6^i-1177/5+5n) = 5
a((2417/5)*6^i-1177/5+5n+1) = (7251/5)*6^i - 2046/5 + 3n
a((2417/5)*6^i-1177/5+5n+2) = 3
a((2417/5)*6^i-1177/5+5n+3) = (2417/5)*6^i - 1162/5 + 5n
a((2417/5)*6^i-1177/5+5n+4) = (7251/5)*6^i - 2041/5 + 3n.

A283885 Relative of Hofstadter Q-sequence: a(n) = max(0, n+3442) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

Original entry on oeis.org

6, 3443, 3444, 3445, 9, 3446, 3447, 3448, 12, 3449, 3450, 3451, 15, 3452, 3453, 17, 3455, 18, 3455, 3457, 3458, 22, 21, 6895, 6889, 9, 18, 6904, 6907, 3451, 22, 3472, 3477, 3455, 27, 36, 3479, 6894, 3446, 39, 3480, 3486, 3450, 42

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 3442 terms.
Most terms in this sequence appear in long period-5 quasilinear runs. These runs are separated by 11943 other terms, and each run is approximately six times as long as the previous.
The first such run that falls into a predictable pattern begins at index 90682, though there are other similar patterns earlier.

Nathan Fox, Table of n, a(n) for n = 1..100000

Cf. A005185, A267501, A274058, A278055, A278066, A283884, A283886, A283887, A283888.

A283885:=proc(n) option remember: if n <= 0 then max(0, n+3442): else A283885(n-A283885(n-1)) + A283885(n-A283885(n-2)) + A283885(n-A283885(n-3)): fi: end:

If the index is between 67 and 3443 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+3444, a(7n+2) = 7n+3446, a(7n+3) = 7, a(7n+4) = 2n+6929, a(7n+5) = n+6877, a(7n+6) = 3440.
For nonnegative integers i, if 1 <= 5n + r <= (487329/5)*6^(i+1) - 88639/5, then
a((487329/5)*6^i - 28924/5 + 5n) = 5
a((487329/5)*6^i - 28924/5 + 5n + 1) = (1461987/5)*6^i - 52797/5 + 3n
a((487329/5)*6^i - 28924/5 + 5n + 2) = 3
a((487329/5)*6^i - 28924/5 + 5n + 3) = (487329/5)*6^i - 28909/5 + 5n
a((487329/5)*6^i - 28924/5 + 5n + 4) = (1461987/5)*6^i - 52792/5 + 3n.

A283886 Relative of Hofstadter Q-sequence: a(n) = max(0, n+19395) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

Original entry on oeis.org

6, 19396, 19397, 19398, 9, 19399, 19400, 19401, 12, 19402, 19403, 19404, 15, 19405, 19406, 17, 19408, 18, 19408, 19410, 19411, 22, 21, 38801, 38795, 9, 18, 38810, 38813, 19404, 22, 19425, 19430, 19408, 27, 36, 19432, 38800, 19399, 39

Offset: 1

Nathan Fox, Mar 19 2017

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 19395 terms.
Most terms in this sequence appear in a long pattern stretching from a(58340) through a(80425266), of 16 interleaved sequences.
This sequence has exactly 80425397 terms (of positive index). a(80425397) = 0, so an attempt to calculate a(80425398) would refer to itself.

Nathan Fox, Table of n, a(n) for n = 1..61000

Cf. A005185, A267501, A274058, A278055, A278066, A283884, A283885, A283887, A283888.

A283886:=proc(n) option remember: if n <= 0 then max(0, n+19395): else A283886(n-A283886(n-1)) + A283886(n-A283886(n-2)) + A283886(n-A283886(n-3)): fi: end:

If the index is between 67 and 19396 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+19397, a(7n+2) = 7n+19399, a(7n+3) = 7, a(7n+4) = 2n+38835, a(7n+5) = n+38783, a(7n+6) = 19393.

A283887 Relative of Hofstadter Q-sequence: a(n) = max(0, n+20830) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

Original entry on oeis.org

6, 20831, 20832, 20833, 9, 20834, 20835, 20836, 12, 20837, 20838, 20839, 15, 20840, 20841, 17, 20843, 18, 20843, 20845, 20846, 22, 21, 41671, 41665, 9, 18, 41680, 41683, 20839, 22, 20860, 20865, 20843, 27, 36, 20867, 41670, 20834, 39

Offset: 1

Nathan Fox, Mar 19 2017

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 20830 terms.
Most terms in this sequence appear in one of two long patterns of 16 interleaved sequences. The first stretches from a(64180) through a(9029945). The second stretches from a(9029971) through a(-20830 + 84975*2^560362).
This sequence has exactly -20799 + 84975*2^560362 terms (of positive index). a(-20799 + 84975*2^560362) = 0, so an attempt to calculate a(-20798 + 84975*2^560362) would refer to itself.

Nathan Fox, Table of n, a(n) for n = 1..68000

Cf. A005185, A267501, A274058, A278055, A278066, A283884, A283885, A283886, A283888.

A283887:=proc(n) option remember: if n <= 0 then max(0, n+20830): else A283887(n-A283887(n-1)) + A283887(n-A283887(n-2)) + A283887(n-A283887(n-3)): fi: end:

If the index is between 67 and 20831 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+20832, a(7n+2) = 7n+20834, a(7n+3) = 7, a(7n+4) = 2n+41705, a(7n+5) = n+41653, a(7n+6) = 20828.

A283888 Relative of Hofstadter Q-sequence: a(n) = max(0, n+27298) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

Original entry on oeis.org

6, 27299, 27300, 27301, 9, 27302, 27303, 27304, 12, 27305, 27306, 27307, 15, 27308, 27309, 17, 27311, 18, 27311, 27313, 27314, 22, 21, 54607, 54601, 9, 18, 54616, 54619, 27307, 22, 27328, 27333, 27311, 27, 36, 27335, 54606, 27302, 39

Offset: 1

Nathan Fox, Mar 19 2017

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 27298 terms.
Most terms in this sequence appear in a long pattern stretching from a(85652) through a(141867984), of 16 interleaved sequences.
This sequence has exactly 141868181 terms (of positive index). a(141868181) = 0, so an attempt to calculate a(141868182) would refer to itself.

Nathan Fox, Table of n, a(n) for n = 1..90000

Cf. A005185, A267501, A274058, A278055, A278066, A283884, A283885, A283886, A283887.

A283888:=proc(n) option remember: if n <= 0 then max(0, n+27298): else A283888(n-A283888(n-1)) + A283888(n-A283888(n-2)) + A283888(n-A283888(n-3)): fi: end:

If the index is between 67 and 27299 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+27300, a(7n+2) = 7n+27302, a(7n+3) = 7, a(7n+4) = 2n+54641, a(7n+5) = n+54589, a(7n+6) = 27296.

A283893 Relative of Hofstadter Q-sequence: a(1) = 3, a(2) = 2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

Original entry on oeis.org

3, 2, 3, 5, 2, 8, 2, 8, 5, 4, 10, 10, 6, 13, 7, 8, 9, 9, 8, 20, 9, 8, 20, 13, 12, 19, 15, 11, 22, 16, 12, 28, 11, 28, 15, 17, 28, 13, 29, 25, 18, 22, 21, 28, 22, 22, 24, 32, 21, 31, 36, 17, 26, 39, 30, 28, 37, 25, 36, 28, 23, 57, 33, 14, 64, 19, 35, 32, 56, 26, 29, 43, 44, 28, 36

Offset: 1

Nathan Fox, Mar 19 2017

nonn

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

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

Cf. A005185, A278066, A278067, A278068, A283894, A283895, A283896, A283897.

A283893:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 3: elif n = 2 then 2: else A283893(n-A283893(n-1)) + A283893(n-A283893(n-2)): fi: end:

A283894 Relative of Hofstadter Q-sequence: a(1) = 17, a(2) = 2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

Original entry on oeis.org

17, 2, 17, 2, 17, 2, 17, 2, 17, 2, 17, 2, 17, 2, 17, 2, 17, 19, 2, 36, 2, 36, 2, 36, 2, 36, 2, 36, 2, 36, 2, 36, 2, 36, 2, 36, 19, 4, 38, 38, 34, 4, 55, 38, 17, 4, 91, 38, 17, 4, 127, 38, 17, 21, 40, 4, 34, 57, 4, 21, 72, 34, 2, 70, 2, 70, 2, 70, 2, 70, 19, 19, 42, 76, 2

Offset: 1

Nathan Fox, Mar 19 2017

nonn

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

Most terms in this sequence alternate between 2 and a large number. Such runs of terms are separated by 102 other terms, and each run is approximately twice as long as the previous.

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

Cf. A005185, A278066, A278067, A278068, A283893, A283895, A283896, A283897.

A283894:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 17: elif n = 2 then 2: else A283894(n-A283894(n-1)) + A283894(n-A283894(n-2)): fi: end:

For nonnegative integers i, if 1 <= 2n + r <= 594*2^(i+1) - 2, then
a(594*2^i + 100 + 2n) = 2
a(594*2^i + 100 + 2n + 1) = 594*2^(i+1) - 2.

A283895 Relative of Hofstadter Q-sequence: a(1) = 41, a(2) = 2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

Original entry on oeis.org

41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 2, 41, 43, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2, 84, 2

Offset: 1

Nathan Fox, Mar 19 2017

nonn

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

Most terms in this sequence alternate between 2 and a large constant. Such runs of terms are eventually separated by either 224 or 562 other terms, and each run is approximately twice as long as the previous.

Nathan Fox, Table of n, a(n) for n = 1..50000

Cf. A005185, A278066, A278067, A278068, A283893, A283894, A283896, A283897.

A283895:=proc(n) option remember: if n <= 0 then 0: elif n = 1 then 41: elif n = 2 then 2: else A283895(n-A283895(n-1)) + A283895(n-A283895(n-2)): fi: end:

cp's OEIS Frontend

A278067 Relative of Hofstadter Q-sequence: a(1) = 37, a(2) = 2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

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A278068 Relative of Hofstadter Q-sequence: a(1) = 57, a(2) = 2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

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A283884 Relative of Hofstadter Q-sequence: a(n) = max(0, n+193) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

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A283885 Relative of Hofstadter Q-sequence: a(n) = max(0, n+3442) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

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A283886 Relative of Hofstadter Q-sequence: a(n) = max(0, n+19395) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

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A283887 Relative of Hofstadter Q-sequence: a(n) = max(0, n+20830) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

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A283888 Relative of Hofstadter Q-sequence: a(n) = max(0, n+27298) for n <= 0; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 0.

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A283893 Relative of Hofstadter Q-sequence: a(1) = 3, a(2) = 2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

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Maple

A283894 Relative of Hofstadter Q-sequence: a(1) = 17, a(2) = 2; thereafter a(n) = a(n-a(n-1)) + a(n-a(n-2)).

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