A267501 -id:A267501 - 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.

A274058 Relative of Hofstadter Q-sequence: a(n) = max(0, n+32478) 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, 32479, 32480, 32481, 9, 32482, 32483, 32484, 12, 32485, 32486, 32487, 15, 32488, 32489, 17, 32491, 18, 32491, 32493, 32494, 22, 21, 64967, 64961, 9, 18, 64976, 64979, 32487, 22, 32508, 32513, 32491, 27, 36, 32515, 64966, 32482, 39, 32516, 32522

Offset: 1

Nathan Fox, Nov 10 2016

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 32478 terms.
This sequence has exactly 37025 terms (of positive index).  a(37025) = 0, so an attempt to calculate a(37026) would refer to itself.
Without the convention that a(n) = 0 for n <= -32478, this sequence would have exactly 24 terms (of positive index), since computing a(25) refers to a(-64942).
If 32478 in this sequence's definition is replaced by any larger number congruent to 5 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..37025
N. Fox, Hofstadter-like Sequences over Nonstandard Integers", Talk given at the Rutgers Experimental Mathematics Seminar, November 10 2016.

Cf. A005185, A267501, A278055, A373234, A373235, A373236, A373237, A373238, A373239.

a[n_] := a[n] = If[n <= 0, Max[0, n + 2^15 - 290], a[n - a[n - 1]] + a[n - a[n - 2]] + a[n - a[n - 3]]]; Array[a, 42] (* Robert G. Wilson v, Mar 19 2017 *)

If the index is between 67 and 32479 (inclusive), then a(7n) = 7n+2, a(7n+1) = 7n+32480, a(7n+2) = 7n+32482, a(7n+3) = 7, a(7n+4) = 2n+65001, a(7n+5) = n+64949, a(7n+6) = 32476.

Formula and definition corrected by Nathan Fox, Mar 18 2017

A267502 Number of cycles of length 3 of autobiographical numbers (A267491 ... A267498) in base n.

Original entry on oeis.org

0, 3, 0, 0, 0, 3, 9, 18, 45

Offset: 2

Antonia Münchenbach, Jan 28 2016

a(n) is the number of cycles of length 3 of autobiographical numbers in base n. For n>=5, it seems that a(n)=3/2*n^2-33/2*n+45 describes the number of cycles of length 3 in base n. The formula is correct for 5<=n<=11, but unknown for n>11. We assume it's correct for all n>=5.

			In base two, four, five and six there is no cycle of length 3.
In base three, there is 1 cycle of length 3 with 3 numbers:  10011112, 10101102, 2012112.
In base 10, there are 6 cycles of length 3 (18 numbers).

Antonia Münchenbach and Nicole Anton George, "Eine Abwandlung der Conway-Folge", contribution to "Jugend forscht" 2016, 2016

Cf. A047841, A267491, A267492, A267493, A267494, A267495, A267496, A267497, A267498, A267499, A267500, A267501, A267502.

Conjecture: a(n) = 3/2*n^2 - 33/2*n + 45. The formula is correct for 5<=n<=11, but unknown for n>11. We assume it's correct for all n>=5.

A268368 An eventually quasi-quadratic sequence satisfying a Hofstadter-like recurrence.

Original entry on oeis.org

0, 1, 0, 4, 4, 4, 3, 12, 8, 4, 3, 24, 12, 4, 3, 40, 16, 4, 3, 60, 20, 4, 3, 84, 24, 4, 3, 112, 28, 4, 3, 144, 32, 4, 3, 180, 36, 4, 3, 220, 40, 4, 3, 264, 44, 4, 3, 312, 48, 4, 3, 364, 52, 4, 3, 420, 56, 4, 3, 480, 60, 4, 3, 544, 64, 4, 3, 612, 68, 4, 3, 684

Offset: 1

Nathan Fox, Feb 23 2016

a(n) is the solution to the recurrence relation a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)), with the initial conditions: a(n) = 0 if n <= 0; a(1) = 0, a(2) = 1, a(3) = 0, a(4) = 4, a(5) = 4, a(6) = 4, a(7) = 3.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A005185, A188670, A244477, A264757, A267501.

concat(0, Vec(x^2*(1 + 4*x^2 + 4*x^3 + x^4 + 3*x^5 - 4*x^7 - 5*x^8 - 6*x^9 + 3*x^12 + 3*x^13) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3) + O(x^100))) \\ Colin Barker, Jun 22 2017

a(2) = 1, a(3) = 0; otherwise a(4n) = 2n^2+2n, a(4n+1) = 4n, a(4n+2) = 4, a(4n+3) = 3.
From Colin Barker, Jun 22 2017: (Start)
G.f.: x^2*(1 + 4*x^2 + 4*x^3 + x^4 + 3*x^5 - 4*x^7 - 5*x^8 - 6*x^9 + 3*x^12 + 3*x^13) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n>12.
(End)

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.

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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A274058 Relative of Hofstadter Q-sequence: a(n) = max(0, n+32478) 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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A267502 Number of cycles of length 3 of autobiographical numbers (A267491 ... A267498) in base n.

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A268368 An eventually quasi-quadratic sequence satisfying a Hofstadter-like recurrence.

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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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