A272727 -id:A272727 - cp's OEIS frontend

A272729 a(n) is the number of repetitions of 2n-1 in A272727.

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

Offset: 1

Ivan Neretin, May 05 2016

nonn

Also, value of A272728 at the n-th local maximum.
Also, the trajectory of 1 under the morphism n->[1,1..1,n+1] (the number of 1's is n-1).
Average value tends to 2.
Number n makes its first appearance at the position 2^(n-1) and has frequency 1/2^n.
Conjectured first differences of A037988 (true for at least 8192 terms). - Sean A. Irvine, Jun 26 2022

			The morphism acts as follows:
1->2; 2->1,3; 3->1,1,4; 4->1,1,1,5; etc.
The trajectory starts as:
1 ->
2 ->
1,3 ->
2,1,1,4 ->
1,3,2,2,1,1,1,5 -> ...
The result of k iterations is a series with 2^(k-1) terms; their sum is 2^k.
If A001511 is laid out in a similar irregular triangle, each row
would contain the same terms, albeit in a different order:
1,
2,
1,3,
1,2,1,4,
1,2,1,3,1,2,1,5...

Cf. A001511, A272727, A272728.

Flatten@NestList[Flatten[Append[ConstantArray[1, # - 1], # + 1] & /@ #] &, {1}, 7]

A272728 a(n) = A272727(2n-1) - n.

Original entry on oeis.org

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

Offset: 1

Ivan Neretin, May 05 2016

nonn

a(n)>=0.
|a(n+1)-a(n)|=1.
All local minima occur where a(n)=0. Values at the local maxima are listed in A272729.
Empirically: The least n such that a(n) = k - 1 is n = 2^k - k. - Danny Rorabaugh, May 12 2016

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A272727, A272729.

nn = 120; s = Nest[Append[#, Count[# - Reverse[#], x_ /; x == 0]] &, {0}, 2 nn - 1]; Table[s[[2 n]] - n, {n, nn}] (* Michael De Vlieger, May 05 2016, after Ivan Neretin at A272727 *)

A379266 a(n) is the number of coincidences of the first n terms of this sequence and the first n terms of A379265 in reverse order, i.e., the number of equalities a(k) = A379265(n-1-k) for 0 <= k < n.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 1, 2, 1, 0, 3, 1, 0, 2, 0, 2, 2, 2, 2, 3, 3, 3, 1, 0, 3, 2, 3, 3, 4, 5, 4, 4, 4, 3, 3, 3, 2, 2, 2, 3, 6, 6, 6, 6, 8, 7, 6, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 3, 4, 2, 1, 0, 5, 4, 4, 5, 6, 7, 8, 7, 9, 10, 11, 12, 12, 13, 16, 16, 16, 16, 14, 12

Offset: 0

Pontus von Brömssen, Dec 19 2024

nonn

a(n) appears to grow roughly like sqrt(n).

Pontus von Brömssen, Table of n, a(n) for n = 0..9999
Pontus von Brömssen, Plot of A379265(n), A379266(n), and sqrt(n) for n=0..100000.

Cf. A272727, A379265, A379297.

def A379266_list(nterms):
    A = []
    A379265 = []
    for n in range(nterms):
        a = sum(1 for x, y in zip(A, reversed(A379265)) if x==y)
        if n != 0:
            b += (b==A[-1])
        else:
            b = 0
        A.append(a)
        A379265.append(b)
    return A

A379265 a(n) is the number of coincidences of the first n terms of this sequence and A379266, i.e., the number of equalities a(k) = A379266(k) for 0 <= k < n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 10, 11, 12, 13, 13, 14, 14, 14, 14, 14, 15

Offset: 0

Pontus von Brömssen, Dec 19 2024

nonn

a(n) appears to grow roughly like sqrt(n).

Pontus von Brömssen, Table of n, a(n) for n = 0..9999
Pontus von Brömssen, Plot of A379265(n), A379266(n), and sqrt(n) for n=0..100000.

Cf. A272727, A379266, A379297.

def A379265_list(nterms):
    A = []
    A379266 = []
    for n in range(nterms):
        if n != 0:
            a += (a==A379266[-1])
        else:
            a = 0
        b = sum(1 for x,y in zip(A,reversed(A379266)) if x==y)
        A.append(a)
        A379266.append(b)
    return A

For n >= 1, a(n) = a(n-1)+1 if a(n-1) = A379266(n-1), otherwise a(n) = a(n-1).

A276638 a(0)=0; thereafter a(n) is the number of matches between the first n terms circularly shifted by s and the reverse of the first n terms, maximized over s.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 4, 3, 6, 3, 6, 3, 8, 5, 8, 5, 6, 5, 6, 7, 8, 7, 10, 7, 8, 7, 12, 7, 12, 7, 12, 7, 12, 7, 14, 9, 14, 9, 14, 9, 14, 9, 14, 9, 14, 11, 16, 11, 14, 11, 16, 11, 18, 11, 16, 11, 16, 11, 16, 11, 20, 13, 16, 13, 18, 13, 16, 13, 18, 13, 18, 13, 18

Offset: 0

Andrey Zabolotskiy, Sep 08 2016

a(n) is the number of matches between (a(s), ..., a(n-1), a(0), ..., a(s-1)) and (a(n-1), ..., a(0)), maximized over s.
The odd bisection of the sequence (i. e., the subsequence a(2k+1)) appears to be bound both above and below by n^0.63 asymptotically. It includes odd terms only and grows monotonically with many plateaus.
The even bisection of the sequence (i. e., the subsequence a(2k)) appears to be bound both above and below asymptotically by the same power function as the odd bisection with larger coefficients. However, its behavior differs in other aspects: it includes even terms only and exhibits stochastic oscillations with increasing amplitude.

			The first 6 terms (0, 1, 2, 1, 4, 3) shifted by 5 to the left yield (3, 0, 1, 2, 1, 4), which coincides with the first 6 terms reversed (3, 4, 1, 2, 1, 0) at 4 positions, and no shift produces more matches than 4, thus a(6)=4.

Andrey Zabolotskiy, Table of n, a(n) for n = 0..9999

Cf. A272727.

a = {0}; Do[AppendTo[a, Max@ Map[Count[Transpose@ #, w_ /; Equal @@ w] &, Array[{RotateLeft[a, #], Reverse@ a} &, n]]], {n, 72}]; a (* Michael De Vlieger, Sep 13 2016 *)

a = [0]
for n in range(1, 100):
    a.append(max(sum(a[(i+s)%n]==a[-i-1] for i in range(n)) for s in range(n)))

A359807 a(1) = 0; thereafter a(n) is the largest a(i) + i which is < n among i = 1..n-1.

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 4, 7, 7, 9, 10, 11, 11, 11, 11, 15, 16, 16, 16, 19, 19, 21, 21, 23, 24, 25, 26, 26, 26, 26, 26, 31, 31, 33, 34, 35, 35, 35, 35, 39, 40, 40, 40, 43, 44, 44, 44, 47, 47, 49, 49, 51, 51, 53, 54, 55, 56, 57, 57, 57, 57, 57, 57, 63, 64, 64, 64, 67, 67, 69, 69, 71, 72, 73, 74, 74, 74, 74, 74

Offset: 1

Neal Gersh Tolunsky, Jan 13 2023

nonn

Critical values of Conway's game of one-dimensional phutball (A037988), except for the initial zero.
Conjectured run lengths are A272729, and the terms which occur here are partial sums of A272729.
The next distinct term here occurs at index a(i)+i+1 for every index i.

			For n=3, we see that for i=1 and 2, a(i)+i = 1 and 3, of which only 1 is < n=3, so that a(3)=1.
For n=5, i=1..4 have a(i)+i = 1,3,4,7 and the largest < n=5 is 4 so that a(5)=4.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A367039, A037988, A272729, A272727, A094591, A367026.

lista(nn) = my(va = vector(nn)); va[1] = 0; for (n=2, nn, va[n] = vecmax(select(x->(xMichel Marcus, Jan 31 2023

{ my (v = 0, m = 0); for (n = 1, 79, if (bittest(m, n-1), v = n-1); print1 (v", "); m = bitor(m, 2^(v+n))) } \\ Rémy Sigrist, Feb 08 2023

A335060 a(n) is the number of values of k < n for which 4*(a(k) + a(n-k)) < n.

Original entry on oeis.org

0, 1, 0, 2, 2, 1, 2, 0, 4, 4, 2, 5, 6, 4, 2, 3, 10, 8, 6, 2, 6, 9, 6, 4, 8, 8, 8, 4, 12, 9, 10, 7, 10, 14, 10, 6, 8, 14, 14, 11, 12, 8, 16, 10, 14, 10, 16, 15, 14, 14, 16, 14, 12, 10, 16, 11, 18, 16, 14, 22, 20, 12, 12, 17, 24, 18, 22, 18, 22, 16, 10, 19, 26, 23, 18, 22, 24

Offset: 1

Samuel B. Reid, May 21 2020

			a(6) is 1 because, if (a(k) + a(6-k)) * 4 is less than 6, k can only be 3.
a(9) is 4 because k must be 1, 3, 6, or 8 in order for (a(k) + a(9-k)) * 4 to be less than 9.

Samuel B. Reid, Table of n, a(n) for n = 1..10000
Samuel B. Reid, C program for A335060
Samuel B. Reid, Plot of 2000000 terms

Cf. A007306, A272727, A317443.

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lista(nn) = {my(va=vector(nn)); for (n=2, nn, va[n] = sum(k=1, n-1, 4*(va[k] + va[n-k]) < n);); va;} \\ Michel Marcus, May 22 2020

A380188 a(n) is the maximum number of coincidences of the first n terms of this sequence and a cyclic shift of the first n terms of A380189, i.e., the number of equalities a(k) = A380189((s+k) mod n) for 0 <= k < n, maximized over s.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 0

Pontus von Brömssen, Jan 15 2025

nonn

Is a(n+1)-a(n) always 0 or 1?
Consider a pair of sequences b and c defined in the following manner:
  - b(n) is the number of coincidences of the first n terms of sequences b and c,
  - c(n) is the number of coincidences of the first n terms of sequence b and the first n terms of sequence c in reverse order.
(The sequences are "self-starting" with no initial values required, because for n = 0 there are obviously no coincidences, so b(0) = c(0) = 0.) For each of b and c, we may or may not allow circular shifts and maximize the number of coincidences over all such shifts, so there are four versions:
  - No shifts: (b,c) = (A379265,A379266).
  - Shifts in b but not in c: (b,c) = (A380188,A380189).
  - Shifts in c and either shifts or no shifts in b: In both these cases, b and c are the following sequences, which are constant from n = 5 and n = 7, respectively:
      b: 0, 1, 2, 3, 3, 4, 4, 4, 4, 4, ...
      c: 0, 1, 2, 1, 3, 2, 2, 3, 3, 3, ...

			The first time the shift comes into play is for n = 21. The first 21 terms of this sequence and of A380189 are:
  0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
  0, 1, 0, 2, 0, 1, 1, 2, 1, 0, 3, 1, 0, 2, 0, 2, 2, 2, 2, 3, 3
  ^  ^     ^                    ^
with only 4 coincidences. But if the second row is shifted 7 steps to the right, we get:
  0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
  0, 2, 2, 2, 2, 3, 3, 0, 1, 0, 2, 0, 1, 1, 2, 1, 0, 3, 1, 0, 2
  ^     ^  ^     ^  ^
with 5 coincidences. This is the best possible, so a(21) = 5.

Pontus von Brömssen, Table of n, a(n) for n = 0..20000

Cf. A272727, A276638, A379265, A379266, A380189, A380190.

A380189 a(n) is the number of coincidences of the first n terms of this sequence and the first n terms of A380188 in reverse order, i.e., the number of equalities a(k) = A380188(n-1-k) for 0 <= k < n.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 1, 2, 1, 0, 3, 1, 0, 2, 0, 2, 2, 2, 2, 3, 3, 3, 1, 0, 3, 2, 3, 3, 4, 5, 4, 4, 4, 3, 3, 3, 2, 2, 2, 3, 6, 6, 6, 6, 8, 7, 6, 5, 5, 5, 4, 4, 2, 1, 0, 3, 1, 0, 2, 0, 2, 2, 3, 5, 7, 7, 6, 5, 4, 5, 5, 6, 3, 2, 2, 1, 1, 4, 3, 1, 2, 3, 2, 4, 3, 4, 4

Offset: 0

Pontus von Brömssen, Jan 15 2025

			For n = 7, the first 7 terms of this sequence and the first 7 terms of A380188 in reverse order are:
  0, 1, 0, 2, 0, 1, 1
  3, 3, 3, 2, 2, 1, 0
           ^     ^
with 2 coincidences, so a(7) = 2.

Pontus von Brömssen, Table of n, a(n) for n = 0..20000

Cf. A272727, A276638, A379266, A380188, A380190.

A367315 a(1) = 1, and for any n > 0, a(2n) is the number of k's among 1..n such that a(k) <= a(n), a(2n+1) is the number of k's among 1..n such that a(k) >= a(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 1, 5, 2, 6, 1, 7, 2, 8, 1, 4, 9, 10, 1, 7, 7, 12, 1, 5, 13, 14, 1, 9, 10, 16, 1, 6, 17, 14, 6, 19, 1, 20, 1, 7, 21, 19, 5, 20, 6, 24, 1, 8, 25, 18, 10, 27, 1, 28, 1, 9, 29, 26, 6, 28, 5, 32, 1, 10, 33, 22, 14, 35, 1, 34, 4, 23, 17, 38

Offset: 1

Rémy Sigrist, Nov 14 2023

nonn

This sequence is unbounded: for any m > 0, a(4*m) + a(4*m+1) >= 2*m + 1, as both terms are positive, at least one of them must be >= m.
This sequence contains infinitely many 1's: if a(n) > a(k) for all k < n, then a(2*n + 1) = 1, and as the sequence is unbounded, we have infinitely many such n.
This sequence contains all positive integers: for any n > 0, if k is the index of the n-th 1, then a(2*k) = n.

			a(1) = 1 by definition.
a(1) <= a(1) hence a(2) = 1.
a(1) >= a(1) hence a(3) = 1.
a(1) and a(2) <= a(2) hence a(4) = 2.
a(1), a(2), a(3) and a(4) <= a(4) hence a(8) = 4.
a(1), a(2) and a(3) < a(4), a(4) >= a(4) hence a(9) = 1.

Cf. A272727.

{ for (n = 1, #a = vector(76), print1 (a[n] = if (n==1, 1, sum (k=1, n\2, if (n%2==0, a[k] <= a[n\2], a[k] >= a[n\2])))", ")) }

a(2*n) + a(2*n+1) >= n + 1.

cp's OEIS Frontend

A272729 a(n) is the number of repetitions of 2n-1 in A272727.

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Mathematica

A272728 a(n) = A272727(2n-1) - n.

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A379266 a(n) is the number of coincidences of the first n terms of this sequence and the first n terms of A379265 in reverse order, i.e., the number of equalities a(k) = A379265(n-1-k) for 0 <= k < n.

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A379265 a(n) is the number of coincidences of the first n terms of this sequence and A379266, i.e., the number of equalities a(k) = A379266(k) for 0 <= k < n.

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Formula

A276638 a(0)=0; thereafter a(n) is the number of matches between the first n terms circularly shifted by s and the reverse of the first n terms, maximized over s.

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A359807 a(1) = 0; thereafter a(n) is the largest a(i) + i which is < n among i = 1..n-1.

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PARI

PARI

A335060 a(n) is the number of values of k < n for which 4*(a(k) + a(n-k)) < n.

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C

PARI

A380188 a(n) is the maximum number of coincidences of the first n terms of this sequence and a cyclic shift of the first n terms of A380189, i.e., the number of equalities a(k) = A380189((s+k) mod n) for 0 <= k < n, maximized over s.

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A380189 a(n) is the number of coincidences of the first n terms of this sequence and the first n terms of A380188 in reverse order, i.e., the number of equalities a(k) = A380188(n-1-k) for 0 <= k < n.

A367315 a(1) = 1, and for any n > 0, a(2n) is the number of k's among 1..n such that a(k) <= a(n), a(2n+1) is the number of k's among 1..n such that a(k) >= a(n).