A115510 -id:A115510 - cp's OEIS frontend

A352199 a(0)=0, a(1)=1, a(2)=2; thereafter, a(n) is smallest number m not yet in the sequence such that the binary expansions of m and a(n-2) have a 1 in common, but the 1's in m are disjoint from the 1's in a(n-1) and a(n-3).

0, 1, 2, 5, 10, 4, 8, 20, 9, 6, 33, 18, 32, 14, 96, 3, 48, 7, 16, 11, 80, 12, 64, 13, 66, 17, 34, 21, 40, 65, 42, 68, 24, 69, 26, 36, 130, 37, 74, 49, 72, 52, 136, 19, 128, 22, 160, 15, 192, 23, 224, 25, 288, 27, 100, 129, 260, 131, 28, 35, 76, 161, 84, 162, 88

Offset: 0

N. J. A. Sloane, Mar 26 2022

A set-theory analog of A350359. This has the same relationship to A350359 as A115510 does to the EKG sequence A064413, as A252867 does to the Yellowstone permutation A098550, and as A338833 does to the Enots Wolley sequence A336957.

An equivalent definition in terms of sets: S(0) = {}, S(1) = {1}, S(2) = {1,2}; thereafter S(n) is the smallest set (different from the S(i) already defined) of positive integers such that S(n) meets S(n-2) but is disjoint from S(n-1) and S(n-3).

			After a(4) = 10 = 1010_2, a(5) = 4 = 100_2, a(6) = 8 = 1000_2, a(7) must have the form ...?010?_2, and the smallest missing number of that form is 20 = 10100_2 = 20.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A064413, A098550, A115510, A252867, A338833, A350359.

{ s=0; for (n=1, #a=vector(65), if (n<=3, a[n]=n-1, for (v=0, oo, if (!bittest(s,v) && bitand(v,a[n-2]) && !bitand(v,bitor(a[n-3],a[n-1])), a[n]=v; break))); s+=2^a[n]; print1(a[n]", ")) } \\ Rémy Sigrist, Mar 27 2022

A352200 a(0)=0, a(1)=1; thereafter, a(n) is the smallest number m not yet in the sequence such that the binary expansions of m and a(n-1) have a 1 in the same position, but the positions of the 1's in the binary expansions of m and a(n-2) are disjoint, except that the second condition is ignored if it would imply that no choice for m were possible.

Original entry on oeis.org

0, 1, 3, 2, 6, 4, 5, 9, 8, 10, 7, 17, 16, 18, 11, 12, 20, 19, 33, 32, 34, 14, 13, 49, 48, 21, 15, 40, 96, 64, 65, 23, 22, 24, 41, 35, 66, 68, 28, 25, 67, 38, 36, 29, 26, 98, 37, 129, 128, 130, 27, 44, 100, 80, 144, 131, 39, 52, 88, 72, 30, 50, 97, 69, 132, 136, 42, 51, 81, 76, 46, 146, 145, 45, 70, 82, 56, 137, 71, 54, 152, 73, 99, 134, 140, 57

Offset: 0

N. J. A. Sloane, Mar 26 2022

The second condition is ignored precisely when the positions of the 1's in a(n-1) are a subset of the 1's in a(n-2).

This is a set-theory analog of A352187.

			a(0)=0 and a(1)=1=1_2 are given.
a(2) = 3 = 11_2 is disjoint from a(0) and intersects a(1).
a(3) = 2 = 10_2 is disjoint from a(1) and intersects a(2).
Now there is no choice for a(4) that meets both conditions, so we ignore the no-intersection-with-a(n-2) condition, and take a(4) = 6 = 110_2.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Maple program

This completes a set of four pairs of sequences: (A064413 and A115510), (A098550 and A252867), (A336957 and A338833), (A352187 and this sequence, A352200).

Maple
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See link.
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A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

1, 5, 15, 45, 105, 264, 555, 1221, 2445, 4935, 9324, 17941, 32522, 59400, 104808, 184569, 315711, 540540, 902335, 1504800, 2462724, 4014513, 6444425, 10316250, 16283707, 25610886, 39841865, 61720659, 94687230, 144731706, 219282679, 330996105, 495901413, 740046425

Offset: 1

Olivier Gérard, May 31 2024

nonn

The length of each row is given by A000041.
As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.
A069782, A088480, A090107, A090108, A090109, A115510, A115511, A130446, A131717, A132086, A153671, A167904, A296879, A296882, A296885, A296888, A296891, A296894, A296897, A296900, A296903, A296906, A303502, A317945, A335280, A361374.

			Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
.
    1 |  1
    5 |  2  3
   15 |  4  5  6
   45 |  7  8  9 10 11
  105 | 12 13 14 15 16 17 18
  264 | 19 20 21 22 23 24 25 26 27 28 29
  555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
.
The sequence gives the row sums of this triangle.

Cf. A000027, seen as a triangle with shape A000041.
Cf. A373301, the same principle, but starting from integer zero instead of 1.
Cf. A006003, row sums of the integers but for the linear triangle.

Module[{s = 0},
 Table[s +=
   PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -
   s*(s - 1)/2, {n, 1, 30}]]

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A352199 a(0)=0, a(1)=1, a(2)=2; thereafter, a(n) is smallest number m not yet in the sequence such that the binary expansions of m and a(n-2) have a 1 in common, but the 1's in m are disjoint from the 1's in a(n-1) and a(n-3).

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A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

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Mathematica