A081145 -id:A081145 - cp's OEIS frontend

A099004 First differences of A081145.

1, 2, 3, -4, 5, 6, -9, 7, 8, -14, 10, 11, -18, 12, 13, -24, 15, 16, -30, 17, 19, -34, 20, 21, -39, 22, 23, -43, 25, 26, -50, 27, 28, -54, 29, 31, -57, 33, -32, 35, 36, -70, 37, 38, -73, 40, 41, -78, 42, 44, -85, 45, 46, -90, 47, 48, -94, 49, 51, -97, 52, 53, -104, 55, 56, -109, 58, 59, -116, 61, -60, 62, 63

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 17 2003

Does every number appear in the sequence of absolute values (see A308007)?

Chai Wah Wu, Table of n, a(n) for n = 1..5000

Cf. A081145, A081146, A308007, A308021.

f[s_] := Block[{k = 1, d = Abs[Most@s - Rest@s], l = Last@s}, While[MemberQ[s, k] || MemberQ[d, Abs[l - k]], k++ ]; Append[s, k]]; t = Nest[f, {1}, 75]; Rest@t - Most@t (* Robert G. Wilson v, Jun 13 2006 *)

A099004_list, l, s, b1, b2 = [1], 2, 3, set(), set([1])
for n in range(2, 5*10**3+1):
    i = s
    while True:
        m = abs(i-l)
        if not (i in b1 or m in b2):
            A099004_list.append(i-l)
            b1.add(i)
            b2.add(m)
            l = i
            while s in b1:
                b1.remove(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 15 2014

Edited by N. J. A. Sloane, Apr 08 2006

More terms from Robert G. Wilson v, Jun 13 2006

A308009 The terms of A081145 appear to lie on three lines; set a(n) = 0 if A081145(n) is on the lower line, 1 if it is on the middle line, and 2 if it is on the upper line.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2

Offset: 1

N. J. A. Sloane, May 14 2019

nonn

Certainly the first 10^6 terms of A081145 lie on three roughly straight lines, of slopes roughly 0.56, 1.40, 2.24. See the graph of A081145 and the data file of Allan Wilks attached to that entry. It is a strong conjecture that all terms of A081145 lie on three roughly straight lines.
In the first 100000 terms, ignoring the initial three terms, usually we see (2,0,1) repeated many times. Occasionally the pattern will be interrupted when (2,0,1) is followed by one or more copies of (0,1) before the repetitions of (2,0,1) resume. The other transitions 0,0; 0,2; 1,1; 2,1; 2,2 seem to never occur.
It would be nice to have a better understanding of this sequence.

Allan Wilks, Table of n, a(n) for n = 1..100000

Cf. A081145.

A308007 Absolute values of first differences of A081145.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 7, 8, 14, 10, 11, 18, 12, 13, 24, 15, 16, 30, 17, 19, 34, 20, 21, 39, 22, 23, 43, 25, 26, 50, 27, 28, 54, 29, 31, 57, 33, 32, 35, 36, 70, 37, 38, 73, 40, 41, 78, 42, 44, 85, 45, 46, 90, 47, 48, 94, 49, 51, 97, 52, 53, 104, 55, 56, 109, 58, 59, 116, 61, 60, 62, 63

Offset: 1

N. J. A. Sloane, May 13 2019

nonn

The (signed) differences themselves are in A099004, but this sequence is important enough to have its own entry.
Conjectured (see the Slater-Velez and Velez articles) to be a permutation of the positive integers.
It appears that the terms line on two lines (this is true for the first million terms): see A308016-A308020.

Chai Wah Wu, Table of n, a(n) for n = 1..5000
P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, Pacific Journal of Mathematics, Vol. 71, No. 1, 1977, 193-196.
P. J. Slater and W. Y. Velez, Permutations of the Positive Integers with Restrictions on the Sequence of Differences, II, Pacific Journal of Mathematics, Vol. 82, No. 2, 1979, 527-531.
William Y. Velez, Research problems 159-160, Discrete Math., 110 (1992), pp. 301-302.

Cf. A081145, A081146, A099004, A308021.

A308015 List of y-coordinates of successive points on the highest of the three lines formed by the terms of A081145.

Original entry on oeis.org

1, 7, 14, 20, 27, 34, 41, 47, 54, 60, 68, 73, 79, 94, 99, 107, 115, 121, 126, 132, 140, 147, 155, 165, 177, 183, 192, 199, 204, 211, 216, 223, 230, 236, 245, 251, 258, 264, 270, 277, 283, 290, 297, 303, 309, 317, 323, 330, 335, 340, 345, 353, 360, 365, 371

Offset: 1

N. J. A. Sloane, May 15 2019

nonn

See A081145 and A308009 for further information about the three lines.

N. J. A. Sloane, Table of n, a(n) for n = 1..31496

Cf. A081145, A308009, A308010-A308014.

A081146 Inverse permutation to A081145.

Original entry on oeis.org

1, 2, 5, 3, 8, 11, 4, 6, 14, 17, 20, 9, 23, 7, 26, 12, 29, 32, 35, 10, 15, 38, 40, 43, 18, 46, 13, 21, 49, 52, 55, 58, 24, 16, 61, 64, 27, 67, 70, 72, 19, 30, 75, 77, 33, 80, 22, 36, 83, 86, 89, 92, 95, 25, 39, 98, 101, 41, 104, 28, 44, 107, 110, 113, 116, 47, 119, 31, 122

Offset: 1

Don Reble, Mar 08 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A081145.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a081146 = (+ 1) . fromJust . (`elemIndex` a081145_list)
-- Reinhard Zumkeller, Jul 02 2015

f[s_] := Block[{d = Abs[Rest@s - Most@s], k = 1}, While[ MemberQ[d, Abs[k - Last@s]] || MemberQ[s, k], k++ ]; Append[s, k]]; t = Nest[f, {1}, 130]; Table[ Position[t, n], {n, 70}] // Flatten (* Robert G. Wilson v, Jun 09 2006 *)
f[s_] := Block[{k = 1, d = Abs[Most@s - Rest@s], l = Last@s}, While[MemberQ[s, k] || MemberQ[d, Abs[l - k]], k++ ]; Append[s, k]]; t = Nest[f, {1}, 125]; Table[Position[t, n], {n, 71}] // Flatten (* Robert G. Wilson v, Jun 13 2006 *)

A308010 List of x-coordinates of successive points on the lowest of the three lines formed by the terms of A081145.

Original entry on oeis.org

1, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 72, 75, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173, 176

Offset: 1

N. J. A. Sloane, May 15 2019

nonn

See A081145 and A308009 for further information about the three lines.

All of A308010-A308015 and A308017-A308020 show a strange mixture of regular patterns interrupted by irregularities. It would be nice to understand them better.

N. J. A. Sloane, Table of n, a(n) for n = 1..34252

Cf. A081145, A308009, A308011-A308015.

A308011 List of y-coordinates of successive points on the lowest of the three lines formed by the terms of A081145.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 11, 13, 15, 17, 18, 19, 22, 23, 24, 26, 29, 30, 31, 32, 35, 36, 38, 39, 40, 43, 44, 46, 49, 50, 51, 52, 53, 56, 57, 59, 62, 63, 64, 65, 67, 69, 70, 72, 74, 76, 77, 80, 82, 83, 84, 85, 86, 88, 89, 90, 92, 93, 95, 97, 98, 101, 103, 104, 105

Offset: 1

N. J. A. Sloane, May 15 2019

nonn

See A081145 and A308009 for further information about the three lines.

N. J. A. Sloane, Table of n, a(n) for n = 1..34252

Cf. A081145, A308009, A308010-A308015.

A308014 List of x-coordinates of successive points on the highest of the three lines formed by the terms of A081145.

Original entry on oeis.org

1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 74, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178

Offset: 1

N. J. A. Sloane, May 15 2019

nonn

See A081145 and A308009 for further information about the three lines.

N. J. A. Sloane, Table of n, a(n) for n = 1..31496

Cf. A081145, A308009, A308010-A308015.

A308172 a(0)=1; thereafter a(n) = smallest number missing from A081145 after A081145(n) has been calculated.

Original entry on oeis.org

1, 2, 3, 3, 3, 5, 5, 5, 6, 6, 6, 9, 9, 9, 10, 10, 10, 11, 11, 11, 13, 13, 13, 15, 15, 15, 17, 17, 17, 18, 18, 18, 19, 19, 19, 22, 22, 22, 23, 23, 24, 24, 24, 26, 26, 26, 29, 29, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 35, 35, 35, 36, 36, 36, 38, 38, 38, 39, 39, 39, 40, 40, 43

Offset: 0

N. J. A. Sloane, May 16 2019

nonn

More than the usual number of terms are shown in order to emphasize the irregularities.

Usually the terms appear in groups of three, but there are many irregularities. To see them, take this sequence, apply the RUNS transform twice, and bisect the result. This produces a very irregular sequence, 12, 10, 1, 46, 2, 2, 3, 2, 5, 6, 7, 10, 9, 13, 25, 5, 24, 1, 42, ... It would be nice to understand this better.

			n=4: Terms 1 through 4 of A081145 are 1,2,4,7. After A081145(4)=7 has been calculated, the numbers not yet in A081145 are 3,5,6,8,..., so a(4) = 3.

Rémy Sigrist, Table of n, a(n) for n = 0..25000
Rémy Sigrist, PARI program for A308172

Cf. A081145.

PARI
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See Links section.
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A308012 List of x-coordinates of successive points on the middle one of the three lines formed by the terms of A081145.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 73, 76, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168

Offset: 1

N. J. A. Sloane, May 15 2019

nonn

See A081145 and A308009 for further information about the three lines.

N. J. A. Sloane, Table of n, a(n) for n = 1..34254

Cf. A081145, A308009, A308010-A308015.

cp's OEIS Frontend

A099004 First differences of A081145.

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A308009 The terms of A081145 appear to lie on three lines; set a(n) = 0 if A081145(n) is on the lower line, 1 if it is on the middle line, and 2 if it is on the upper line.

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A308007 Absolute values of first differences of A081145.

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A308015 List of y-coordinates of successive points on the highest of the three lines formed by the terms of A081145.

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A081146 Inverse permutation to A081145.

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A308010 List of x-coordinates of successive points on the lowest of the three lines formed by the terms of A081145.

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A308011 List of y-coordinates of successive points on the lowest of the three lines formed by the terms of A081145.

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A308014 List of x-coordinates of successive points on the highest of the three lines formed by the terms of A081145.

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A308172 a(0)=1; thereafter a(n) = smallest number missing from A081145 after A081145(n) has been calculated.

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PARI

A308012 List of x-coordinates of successive points on the middle one of the three lines formed by the terms of A081145.

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