A274058 -id:A274058 - cp's OEIS frontend

A278055 Relative of Hofstadter Q-sequence: a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 5.

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 45, 45, 46, 47, 48, 48, 49, 50, 50, 51

Offset: 1

Nathan Fox, Nov 10 2016

nonn

This sequence is monotonic, with successive terms increasing by 0 or 1. So the sequence hits every positive integer.

A number k appears twice in this sequence if and only if for some i, k is congruent to A057198(i) mod 3^i and k > A057198(i).

Nathan Fox, Table of n, a(n) for n = 1..10000
Nathan Fox, A Slow Relative of Hofstadter's Q-Sequence, arXiv preprint arXiv:1611.08244 [math.NT], 2016.

Cf. A005185, A057198, A063882, A267501, A274058.

a[n_] := a[n] = a[n - a[n -1]] + a[n - a[n -2]] + a[n - a[n -3]]; a[1] = 1; a[2] = 2; a[3] = 3; a[4] = 4; a[5] = 5; Array[a, 71] (* Robert G. Wilson v, Dec 02 2016 *)

A=Vecsmall([]);
a(n)=if(n<7, return(n)); if(#ACharles R Greathouse IV, Nov 19 2016

a(n) ~ 2n/3.

A373234 Relative of Hofstadter Q-sequence: a(n) = max(0, n+196) 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, 197, 198, 199, 9, 200, 201, 202, 12, 203, 204, 205, 15, 206, 207, 17, 209, 18, 209, 211, 212, 22, 21, 403, 397, 9, 18, 412, 415, 205, 22, 226, 231, 209, 27, 36, 233, 402, 200, 39, 234, 240, 204, 42, 236, 243, 16, 235, 243, 223, 40, 235, 55, 416, 212, 46, 245, 256, 25, 38, 58, 835, 406, 200, 61, 71, 455, 394, 194, 72, 268, 270, 7, 457, 395

Offset: 1

Nathan Fox, May 28 2024

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 196 terms.
This sequence has exactly 223 terms (of positive index).  a(223) = 0, so an attempt to calculate a(224) would refer to itself.
Without the convention that a(n) = 0 for n <= -196, this sequence would have exactly 24 terms (of positive index), since computing a(25) refers to a(-378).
If 196 in this sequence's definition is replaced by any larger number congruent to 0 mod 7, the behavior is essentially the same, though the quasilinear part (see Formula section) lasts longer.

Nathan Fox, Table of n, a(n) for n = 1..223

Cf. A005185, A267501, A278055, A373235, A373236, A373237, A373238, A274058, A373239.

If the index is between 67 and 195 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+198, a(7n+2) = 7n+200, a(7n+3) = 7, a(7n+4) = 2n+437, a(7n+5) = n+385, a(7n+6) = 194.

A373235 Relative of Hofstadter Q-sequence: a(n) = max(0, n+2087) 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, 2088, 2089, 2090, 9, 2091, 2092, 2093, 12, 2094, 2095, 2096, 15, 2097, 2098, 17, 2100, 18, 2100, 2102, 2103, 22, 21, 4185, 4179, 9, 18, 4194, 4197, 2096, 22, 2117, 2122, 2100, 27, 36, 2124, 4184, 2091, 39, 2125, 2131, 2095, 42, 2127, 2134, 16, 2126, 2134, 2114, 40, 2126, 55, 4198, 2103, 46, 2136, 2147, 25, 38, 58, 8399, 4188

Offset: 1

Nathan Fox, May 28 2024

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 2087 terms.
This sequence has exactly 2341 terms (of positive index).  a(2341) = 0, so an attempt to calculate a(2342) would refer to itself.
Without the convention that a(n) = 0 for n <= -2087, this sequence would have exactly 24 terms (of positive index), since computing a(25) refers to a(-4790).
If 2087 in this sequence's definition is replaced by any larger number congruent to 1 mod 7, the behavior is essentially the same, though the quasilinear part (see Formula section) lasts longer.

Nathan Fox, Table of n, a(n) for n = 1..2341

Cf. A005185, A267501, A278055, A373234, A373236, A373237, A373238, A274058, A373239.

If the index is between 67 and 2085 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+2089, a(7n+2) = 7n+2091, a(7n+3) = 7, a(7n+4) = 2n+4219, a(7n+5) = n+4167, a(7n+6) = 2085.

A373236 Relative of Hofstadter Q-sequence: a(n) = max(0, n+3201) 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, 3202, 3203, 3204, 9, 3205, 3206, 3207, 12, 3208, 3209, 3210, 15, 3211, 3212, 17, 3214, 18, 3214, 3216, 3217, 22, 21, 6413, 6407, 9, 18, 6422, 6425, 3210, 22, 3231, 3236, 3214, 27, 36, 3238, 6412, 3205, 39, 3239, 3245, 3209, 42, 3241, 3248, 16, 3240, 3248, 3228, 40, 3240, 55, 6426, 3217, 46, 3250, 3261, 25, 38, 58, 12855, 6416, 3205

Offset: 1

Nathan Fox, May 28 2024

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 3201 terms.
This sequence has exactly 3725 terms (of positive index).  a(3725) = 0, so an attempt to calculate a(3726) would refer to itself.
Without the convention that a(n) = 0 for n <= -3201, this sequence would have exactly 24 terms (of positive index), since computing a(25) refers to a(-6388).
If 3201 in this sequence's definition is replaced by any larger number congruent to 2 mod 7, the behavior is essentially the same, though the quasilinear part (see Formula section) lasts longer.

Nathan Fox, Table of n, a(n) for n = 1..3725

Cf. A005185, A267501, A278055, A373234, A373235, A373237, A373238, A274058, A373239.

If the index is between 67 and 3199 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+3203, a(7n+2) = 7n+3205, a(7n+3) = 7, a(7n+4) = 2n+6447, a(7n+5) = n+6395, a(7n+6) = 3199.

A373237 Relative of Hofstadter Q-sequence: a(n) = max(0, n+4315) 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, 4316, 4317, 4318, 9, 4319, 4320, 4321, 12, 4322, 4323, 4324, 15, 4325, 4326, 17, 4328, 18, 4328, 4330, 4331, 22, 21, 8641, 8635, 9, 18, 8650, 8653, 4324, 22, 4345, 4350, 4328, 27, 36, 4352, 8640, 4319, 39, 4353, 4359, 4323, 42, 4355, 4362, 16, 4354, 4362, 4342, 40, 4354, 55, 8654, 4331, 46, 4364, 4375, 25, 38, 58, 17311, 8644

Offset: 1

Nathan Fox, May 28 2024

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 4315 terms.
This sequence has exactly 4875 terms (of positive index).  a(4875) = 0, so an attempt to calculate a(4876) would refer to itself.
Without the convention that a(n) = 0 for n <= -4315, this sequence would have exactly 24 terms (of positive index), since computing a(25) refers to a(-8616).
If 4315 in this sequence's definition is replaced by any larger number congruent to 3 mod 7, the behavior is essentially the same, though the quasilinear part (see Formula section) lasts longer.

Nathan Fox, Table of n, a(n) for n = 1..4875

Cf. A005185, A267501, A278055, A373234, A373235, A373236, A373238, A274058, A373239.

If the index is between 67 and 4313 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+4317, a(7n+2) = 7n+4319, a(7n+3) = 7, a(7n+4) = 2n+8675, a(7n+5) = n+8623, a(7n+6) = 4313.

A373238 Relative of Hofstadter Q-sequence: a(n) = max(0, n+200) 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, 201, 202, 203, 9, 204, 205, 206, 12, 207, 208, 209, 15, 210, 211, 17, 213, 18, 213, 215, 216, 22, 21, 411, 405, 9, 18, 420, 423, 209, 22, 230, 235, 213, 27, 36, 237, 410, 204, 39, 238, 244, 208, 42, 240, 247, 16, 239, 247, 227, 40, 239, 55, 424, 216, 46, 249, 260, 25, 38, 58, 851, 414, 204, 61, 71, 463, 402, 198, 72, 272, 274

Offset: 1

Nathan Fox, May 28 2024

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 200 terms.
This sequence has exactly 220 terms (of positive index).  a(220) = 0, so an attempt to calculate a(221) would refer to itself.
Without the convention that a(n) = 0 for n <= -200, this sequence would have exactly 24 terms (of positive index), since computing a(25) refers to a(-386).
If 200 in this sequence's definition is replaced by any larger number congruent to 4 mod 7, the behavior is essentially the same, though the quasilinear part (see Formula section) lasts longer.

Nathan Fox, Table of n, a(n) for n = 1..220

Cf. A005185, A267501, A278055, A373234, A373235, A373236, A373237, A274058, A373239.

If the index is between 67 and 202 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+202, a(7n+2) = 7n+204, a(7n+3) = 7, a(7n+4) = 2n+445, a(7n+5) = n+393, a(7n+6) = 198.

A373239 Relative of Hofstadter Q-sequence: a(n) = max(0, n+118) 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, 119, 120, 121, 9, 122, 123, 124, 12, 125, 126, 127, 15, 128, 129, 17, 131, 18, 131, 133, 134, 22, 21, 247, 241, 9, 18, 256, 259, 127, 22, 148, 153, 131, 27, 36, 155, 246, 122, 39, 156, 162, 126, 42, 158, 165, 16, 157, 165, 145, 40, 157, 55, 260, 134, 46, 167, 178, 25, 38, 58, 523, 250, 122, 61, 71, 299, 238, 116, 72, 190, 192

Offset: 1

Nathan Fox, May 28 2024

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 118 terms.
This sequence has exactly 127 terms (of positive index). a(127) = 0, so an attempt to calculate a(128) would refer to itself.
Without the convention that a(n) = 0 for n <= -118, this sequence would have exactly 24 terms (of positive index), since computing a(25) refers to a(-222).
If 118 in this sequence's definition is replaced by any larger number congruent to 6 mod 7, the behavior is essentially the same, though the quasilinear part (see Formula section) lasts longer.

Nathan Fox, Table of n, a(n) for n = 1..127

Cf. A005185, A267501, A278055, A373234, A373235, A373236, A373237, A373238, A274058.

If the index is between 67 and 118 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+120, a(7n+2) = 7n+122, a(7n+3) = 7, a(7n+4) = 2n+281, a(7n+5) = n+229, a(7n+6) = 116.

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.

cp's OEIS Frontend

A278055 Relative of Hofstadter Q-sequence: a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 5.

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A373234 Relative of Hofstadter Q-sequence: a(n) = max(0, n+196) 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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A373235 Relative of Hofstadter Q-sequence: a(n) = max(0, n+2087) 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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A373236 Relative of Hofstadter Q-sequence: a(n) = max(0, n+3201) 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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A373237 Relative of Hofstadter Q-sequence: a(n) = max(0, n+4315) 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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A373238 Relative of Hofstadter Q-sequence: a(n) = max(0, n+200) 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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A373239 Relative of Hofstadter Q-sequence: a(n) = max(0, n+118) 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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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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