A229037 -id:A229037 - cp's OEIS frontend

A236336 Lexicographically earliest increasing sequence of positive integers whose graph has no three collinear points.

1, 2, 4, 5, 9, 12, 16, 22, 26, 33, 38, 45, 53, 60, 61, 76, 86, 91, 92, 97, 111, 112, 121, 134, 135, 147, 148, 150, 153, 157, 167, 180, 200, 212, 223, 227, 228, 238, 246, 264, 269, 282, 286, 305, 312, 313, 321, 322, 327, 328, 360, 374, 389, 393, 395, 420, 421

Offset: 1

Tanya Khovanova, Jan 22 2014

nonn

An increasing version of A236335.

			Consider a(5). The previous terms are 1,2,4,5. The value of a(5) can't be 6 because points (3,4),(4,5),(5,6) (corresponding to values a(3),a(4),a(5)) are on the same line: y=x+1. Points (1,1),(3,4),(5,7) are on the same line y=3x/2-1/2, so a(5) can't be 7. Points (2,2),(3,4),(5,8) are on the same line: y=2x-2, so a(5) can't be 8. Thus a(5)=5.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A229037, A185256, A236335.

a:= proc(n) option remember; local i, j, k, ok;
      if n<3 then n
    else for k from 1+a(n-1) do ok:=true;
           for j from n-1 to 2 by -1 while ok do
             for i from j-1 to 1 by -1 while ok do
               ok:= (n-j)*(a(j)-a(i))<>(j-i)*(k-a(j)) od
           od; if ok then return k fi
         od
      fi
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 23 2014

g[1] = 1;
g[n_] := g[n] =
  Min[Complement[Range[g[n - 1] + 1, 500],
    Select[Flatten[
      Table[g[k] + (n - k) (g[j] - g[k])/(j - k), {k, n - 2}, {j,
        k + 1, n - 1}]], IntegerQ[#] &]]]
Table[g[k], {k, 50}]

A248639 Least nonnegative sequence which does not contain a 4-term equidistant subsequence (a(n+k*d); k=0,1,2,3) in arithmetic progression.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 4, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 0, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 2, 2, 0, 2, 2, 2, 4, 4, 3, 0, 0, 0, 2, 0, 1, 2, 0, 4, 2, 1, 5, 0, 2, 1, 1, 2, 1, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 1, 1, 1, 4, 1, 2, 3, 0, 1, 2, 1, 0, 3, 3, 4, 1, 1, 3

Offset: 0

M. F. Hasler, Oct 10 2014

nonn

See A248625 for more information, links and examples.

See A248641 for the "positive integers" variant.

Cf. A248625, A248640, A248641, A248627, A229037, A241752.

a=[];for(n=1,190,a=concat(a,0);while(hasAP(a,4),a[#a]++));a \\ See A248625 for hasAP().

A248640 Least nonnegative sequence which does not contain a 5-term equidistant subsequence (a(n+k*d); k=0,1,2,3,4) in arithmetic progression.

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Oct 10 2014

nonn

See A248625 for more information, links and examples.

Cf. A248625, A248639, A248627, A229037, A241752.

a=[];for(n=1,190,a=concat(a,0);while(hasAP(a,5),a[#a]++));a \\ See A248625 for hasAP(). Use concat(a,1) for the "positive integer" variant.

A262942 Sequence of positive integers where each is chosen to be as small as possible subject to the conditions that no three terms a(j), a(j+k), a(j+2*k) (for any j and k) form an arithmetic progression (in any order) and that no term repeats.

Original entry on oeis.org

1, 2, 4, 5, 8, 3, 7, 6, 10, 11, 14, 9, 16, 12, 13, 19, 15, 18, 20, 21, 26, 17, 22, 24, 25, 27, 31, 28, 23, 32, 29, 34, 37, 38, 40, 41, 35, 30, 42, 46, 47, 54, 36, 33, 45, 43, 49, 39, 48, 50, 55, 52, 53, 44, 59, 57, 51, 60, 56, 61, 62, 67, 58, 69, 64, 72, 66, 68, 76, 71, 73, 77, 65, 75, 63, 88, 89, 80, 78, 74, 83, 79, 70, 90, 94, 82, 81, 84, 85, 91, 87, 101

Offset: 1

Max Barrentine, Oct 05 2015

nonn

Conjectured permutation of the natural numbers.

			For n = 4, 3 is not available because {a(2)=2, 3, a(3)=4} form an arithmetic progression, 1,2,4 are already used, so a(4) = 5. - _Robert Israel_, Nov 15 2015

Robert Israel, Table of n, a(n) for n = 1..10000

A229037 has a very similar definition, but a totally different graph.

N:= 1000: # to get all terms before the first > N
V:= Vector(N):
S:= Vector(N):
firstav:= 1;
for n from 1 to N do
    forbid:= {seq(op([2*V[k]-V[2*k-n], 2*V[2*k-n]-V[k],(V[k]+V[2*k-n])/2]),k=ceil((n+1)/2)..n-1)};
    for v from firstav to N do
      if S[v] <> 0 and v = firstav then firstav:= v+1 fi;
      if S[v] = 0 and not member(v, forbid) then
        V[n]:= v;
        S[v]:= 1;
        break
      fi
    od;
    if v > N then break fi;
od:
seq(V[i],i=1..n-1); # Robert Israel, Nov 15 2015

Added more terms from b-file. - N. J. A. Sloane, Nov 26 2015

A293862 Sequence of signed integers where each is chosen to be as small as possible (in absolute value) subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression; in case of a tie, preference is given to the positive value.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, -1, -1, 0, 0, -1, 0, 0, 1, 3, 3, -1, 3, 3, 1, 1, 2, 2, 1, -3, 2, 0, 0, -2, 0, 0, 1, -3, -3, -2, 0, 0, 4, 0, 0, 1, -2, -1, -1, -2, 5, -1, 3, -3, 2, 3, 3, 2, 5, 4, 4, 2, 4, 2, 3, -1, -1, 3, -8, -2, 5, 2, -2, -2, -8, -3, -2, -8, -6, -6, 2, 5

Offset: 1

Rémy Sigrist, Oct 18 2017

sign

This sequence is a "signed" variant of A229037. Graphically, both sequences have similar ethereal features.

For any n > 0, |a(n)| <= floor( (n+1)/4 ).

			a(1) = 0 is suitable.
a(2) = 0 is suitable.
a(3) cannot equal 0 as 2*a(3-1) - a(3-2) = 0.
a(3) = 1 is suitable.
a(4) cannot equal 2 as 2*a(4-1) - a(4-2) = 2.
a(4) = 0 is suitable.
a(5) cannot equal -1 as 2*a(5-1) - a(5-2) = -1.
a(5) cannot equal 2 as 2*a(5-2) - a(5-4) = 2.
a(5) = 0 is suitable.
a(6) cannot equal 0 as 2*a(6-1) - a(6-2) = 0.
a(6) = 1 is suitable.
a(7) cannot equal 2 as 2*a(7-1) - a(7-2) = 2.
a(7) cannot equal -1 as 2*a(7-2) - a(7-4) = -1.
a(7) cannot equal 0 as 2*a(7-3) - a(7-6) = 0.
a(7) = 1 is suitable.
a(8) cannot equal 1 as 2*a(8-1) - a(8-2) = 1.
a(8) cannot equal 2 as 2*a(8-2) - a(8-4) = 2.
a(8) cannot equal 0 as 2*a(8-3) - a(8-6) = 0.
a(8) = -1 is suitable.

Rémy Sigrist, Table of n, a(n) for n = 1..100000
Rémy Sigrist, Scatterplot of the first 1000000 terms
Rémy Sigrist, C++ program for A293862

A333835 Lexicographically earliest sequence of distinct positive integers, which when mapped onto a square spiral, gives a set without three distinct evenly spaced aligned points.

Original entry on oeis.org

1, 2, 3, 4, 17, 18, 20, 21, 22, 24, 27, 28, 31, 33, 34, 61, 80, 81, 87, 90, 93, 100, 131, 135, 145, 146, 148, 152, 154, 157, 158, 160, 166, 171, 172, 174, 189, 194, 225, 253, 268, 270, 271, 276, 281, 282, 291, 294, 295, 298, 316, 335, 338, 368, 383, 397, 405

Offset: 1

Rémy Sigrist, Apr 07 2020

nonn

This sequence has similarities with A005836 and A229037.

			The first terms, mapped onto a square spiral, are:
         *---*---*---*--61---*---*---*---*
         |                               |
         *   *---*---*--34--33---*--31   *
         |   |                       |   |
         *   *  17---*---*---*---*   *   *
         |   |   |               |   |   |
         *   *  18   *---4---3   *   *   *
         |   |   |   |       |   |   |   |
         *   *   *   *   1---2   *  28   *
         |   |   |   |           |   |   |
         *   *  20   *---*---*---*  27   *
         |   |   |                   |   |
         *   *  21--22---*--24---*---*   *
         |   |                           |
         *   *---*---*---*---*---*---*---*
         |
         *---*---*---*---*---*---*--80--81

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of (A174344(a(n)), A274923(a(n))) in the region -10000 <= x, y <= 10000
Rémy Sigrist, C# program for A333835

See A333825 for a similar sequences.

Cf. A005836, A174344, A229037, A274923.

A357256 "Forest Fire" sequence with the additional condition that no progression of the form ABA is allowed for any terms A and B.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 5, 3, 3, 5, 6, 6, 7, 10, 10, 7, 9, 12, 11, 9, 12, 8, 8, 14, 14, 11, 15, 13, 13, 17, 23, 20, 16, 15, 17, 23, 24, 16, 18, 18, 19, 26, 21, 28, 25, 19, 22, 22, 29, 24, 20, 30, 27, 21, 32, 29, 30, 35, 26, 34, 36, 25, 31, 32, 34, 37, 39, 36, 28, 27

Offset: 1

Neal Gersh Tolunsky, Dec 11 2022

nonn

It is easy to see that a number can occur no more than twice: 1) If a number occurs twice, one term with that value must be at an odd n and the other at an even n. This is because otherwise you could always find a progression of the form ABA. 2) Once two terms of the same value are in the sequence on an even and odd n, no third term with that value can be added without creating a progression of form ABA.

			a(4)=2 because if a(4) were 1 the 2-4th terms would be the ABA-form progression 1,2,1. 2 here is the smallest number which forms neither an arithmetic nor ABA progression.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A229037.

from itertools import count, islice
def agen(): # generator of terms
    alst, mink, aba = [0], [1, 1], [set(), set()] # even, odd appearances
    for n in count(1):
        k = mink[n&1]
        ff = set(2*alst[n-i] - alst[n-2*i] for i in range(1, (n+1)//2))
        while k in ff or k in aba[n&1]: k += 1
        alst.append(k); aba[n&1].add(k); yield k
        while mink[n&1] in aba[n&1]: mink[n&1] += 1
print(list(islice(agen(), 70))) # Michael S. Branicky, Dec 12 2022

More terms from Michael S. Branicky, Dec 12 2022

A361702 Lexicographically earliest sequence of positive numbers on a square spiral such that no four equal numbers lie on the circumference of a circle.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 3, 1, 3, 3, 3, 2, 2, 3, 2, 4, 4, 4, 2, 1, 2, 3, 4, 3, 4, 4, 5, 5, 5, 5, 1, 1, 5, 4, 3, 4, 6, 5, 6, 6, 4, 3, 2, 1, 5, 4, 1, 6, 3, 4, 2, 5, 6, 5, 6, 7, 6, 7, 3, 1, 5, 7, 7, 6, 4, 6, 5, 7, 6, 4, 7, 8, 7, 6, 7, 4, 7, 5, 8, 8, 8, 6, 3, 6, 4, 8, 5, 8, 9, 9, 7, 8, 3

Offset: 1

Scott R. Shannon, Mar 21 2023

nonn

The first term a(1) = 1 lies at the (0,0) origin while all other terms lie on integer coordinates.

			a(4) = 2 as a(1) = a(2) = a(3) = 1 all lie on the circumference of a circle with radius 1/sqrt(2) centered at (1/2,1/2), assuming a counter-clockwise spiral, so a(4) cannot be 1.
a(12) = 3 as a(2) = a(3) = a(11) = 1 all lie on the circumference of a circle with radius 1/sqrt(2) centered at (3/2,1/2), so a(12) cannot be 1, while a(4) = a(8) = a(10) = 2 all lie on the circumference of a circle with radius sqrt(2) centered at (1,0), so a(12) cannot be 2.
a(22) = 4 as a(1) = a(2) = a(7) = 1 all lie on the circumference of a circle with radius sqrt(10)/2 centered at (1/2,-3/2), so a(22) cannot be 1, a(6) = a(19) = a(21) = 2 all lie on the circumference of a circle with radius sqrt(5)/2 centered at (-3/2,-1), so a(22) cannot be 2, while a(12) = a(16) = a(20) = 3 all lie on the circumference of a circle with radius sqrt(5) centered at (0,0), so a(22) cannot be 3.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 10000 terms on the square spiral. The values are scaled across the spectrum from red to violet to show their relative size. Zoom in to see the numbers.
Scott R. Shannon, Image highlighting the 1 valued terms in the first 10000 terms on the square spiral. Zoom in to see the numbers.

Cf. A361486, A274640, A229037, A174344, A274923, A346294.

A366574 a(1) = 1; for n > 1, a(n) is the maximum positive k such that all terms a(t), a(t-m), a(t-2m), ..., a(t-(k-1)m), for 0=1, are equal.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 3, 1, 2, 3, 2, 3, 2, 3, 3, 3, 4, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 5, 1, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 3, 4, 5, 3, 4, 5, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 5, 5, 6, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 3, 4, 5, 6, 3, 4, 5, 6, 3, 4, 5, 6, 4

Offset: 1

Scott R. Shannon, Oct 13 2023

nonn

The terms form quickly form a repetitive pattern of arithmetic progressions of increasing length, see the graph. This leads to any given value t eventually being in a progression of length t+1 which then never increases.

See A366724 for the index where a number first appears.

			a(3) = 2 as a(2) = 1 and a(2) = a(1) = 1.
a(11) = 3 as a(10) = 2 and a(7) = a(6) = a(5) = 2.
a(18) = 4 as a(17) = 3 and a(17) = a(15) = a(13) = a(11) = 3.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A366724, A178976, A308638, A229037.

A367196 Lexicographically earliest sequence such that for any distinct j, k, m that are the side lengths of a triangle, a(j), a(k), and a(m) are not the side lengths of a triangle.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 5, 1, 8, 13, 21, 2, 34, 55, 89, 1, 144, 233, 4, 377, 610, 987, 1597, 1, 17, 2584, 4181, 6765, 10946, 17711, 3, 72, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 1, 7, 305, 832040, 1346269, 2178309, 3524578, 41, 5702887, 1292, 9227465

Offset: 1

Neal Gersh Tolunsky, Nov 09 2023

nonn

In a triangle, the sum of any two side lengths is greater than that of the third, so that x + y > z. The empty triangle (or line) is not counted, which means that x + y cannot be equal to z. In practice, if we have two side lengths x and y, we can find their sum s and their difference d, which tells us that side z must fall in the range d < z < s to form a triangle.
For n>0, A002620(n+1) gives the number of combinations of three indices whose corresponding terms cannot be the side lengths of a triangle in this sequence.
It appears that the local maxima are the Fibonacci numbers A000045 (except for 1s).
The second-largest values in the log graph, falling roughly on a line, appear to be A001076 (half of the even Fibonacci numbers).
Generalizing the sequence to prohibit the side lengths of any n-gon at distinct n-gonal indices gives A011782.

			a(3)=1 because the indices 1,2,3 could not be the side lengths of a triangle, so there is no restriction and the smallest number is chosen.
a(7) cannot be 1 because a(3)=1, a(5)=1, and a(7)=1 could be the side lengths of a triangle at indices which are also side lengths of a triangle.
a(7) cannot be 2 because a(4)=2, a(6)=3, and a(7)=2 are side lengths of a triangle at indices that forbid it.
a(7) cannot be 3 because a(5)=1, a(6)=3, and a(7)=3 also make a triangle at indices that forbid it.
a(7) cannot be 4 because a(4)=2, a(6)=3 and a(7)=4 form a triangle at unsuitable indices.
a(7) can be 5 without contradiction, so a(7)=5.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..2000
Samuel Harkness, MATLAB program

Cf. A000045, A001076.
Cf. A229037, A276204, A364057, A361933.
Cf. A316841, A070080 (triangle side lengths).

MATLAB
```
See Links.
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a(11)-a(50) from Samuel Harkness, Nov 13 2023

cp's OEIS Frontend

A236336 Lexicographically earliest increasing sequence of positive integers whose graph has no three collinear points.

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A248639 Least nonnegative sequence which does not contain a 4-term equidistant subsequence (a(n+k*d); k=0,1,2,3) in arithmetic progression.

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A248640 Least nonnegative sequence which does not contain a 5-term equidistant subsequence (a(n+k*d); k=0,1,2,3,4) in arithmetic progression.

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A262942 Sequence of positive integers where each is chosen to be as small as possible subject to the conditions that no three terms a(j), a(j+k), a(j+2*k) (for any j and k) form an arithmetic progression (in any order) and that no term repeats.

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A293862 Sequence of signed integers where each is chosen to be as small as possible (in absolute value) subject to the condition that no three terms a(j), a(j+k), a(j+2k) (for any j and k) form an arithmetic progression; in case of a tie, preference is given to the positive value.

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A333835 Lexicographically earliest sequence of distinct positive integers, which when mapped onto a square spiral, gives a set without three distinct evenly spaced aligned points.

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A357256 "Forest Fire" sequence with the additional condition that no progression of the form ABA is allowed for any terms A and B.

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Python

Extensions

A361702 Lexicographically earliest sequence of positive numbers on a square spiral such that no four equal numbers lie on the circumference of a circle.

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A366574 a(1) = 1; for n > 1, a(n) is the maximum positive k such that all terms a(t), a(t-m), a(t-2*m), ..., a(t-(k-1)*m), for 0=1, are equal.

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A366574 a(1) = 1; for n > 1, a(n) is the maximum positive k such that all terms a(t), a(t-m), a(t-2m), ..., a(t-(k-1)m), for 0=1, are equal.