A349526 -id:A349526 - cp's OEIS frontend

A349946 a(n) = A349526(n) + A349526(n+1).

2, 3, 4, 3, 4, 5, 5, 6, 4, 5, 6, 6, 7, 7, 8, 5, 6, 7, 7, 8, 8, 9, 9, 10, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15

Offset: 1

Clark Kimberling, Dec 07 2021

nonn

Every positive integer n >= 2 occurs exactly n-1 times; the last occurrence of n is a((n-1)^2).

			A349426 = (1,1,2,2,1,3,2,3,3,1,4,2,4,3,4,4,1,...), in which every pair i,j of positive integers occurs exactly once; a(1) = 1+1, a(2) = 1+2, a(3) = 2+2.

Cf. A349526, A349947 (array: row n shows positions of n+1 in A349946).

t = {1, 1}; Do[t = Join[t, Riffle[Range[n], n], {n}], {n, 2, 100}];
u = Flatten[Partition[t, 2]];
v = Table[n (n + 1), {n, 1, 80}];
d = Delete[u, Map[{#} &, v]]; (* A349526 *)
p = Table[{d[[n]], d[[n + 1]]}, {n, 1, 150}];
Map[Total, p]  (* A349946 *)

A349947 Triangular array: row n gives the positions of n+1 in A349946.

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 6, 7, 10, 16, 8, 11, 12, 17, 25, 13, 14, 18, 19, 26, 36, 15, 20, 21, 27, 28, 37, 49, 22, 23, 29, 30, 38, 39, 50, 64, 24, 31, 32, 40, 41, 51, 52, 65, 81, 33, 34, 42, 43, 53, 54, 66, 67, 82, 100, 35, 44, 45, 55, 56, 68, 69, 83, 84, 101, 121

Offset: 1

Clark Kimberling, Dec 07 2021

Every positive integer occurs exactly once, so as a sequence, this is a permutation of the positive integers.
Row n ends in n^2. The first term in row n is (1 + n/1)^2 - 3 if n >= 4 and n is even; as in A028872(n) for n >= 3.
The first term in row n is ((n+1)/2)^2 - 1 if n >= 3 and n is odd, as in A132411(n) for n >= 3.

			First 7 rows:
   1
   2   4
   3   5   9
   6   7  10  16
   8  11  12  17  25
  13  14  18  19  26  36
  14  20  21  27  28  37  49

Cf. A349526, A349946.

t = {1, 1}; Do[t = Join[t, Riffle[Range[n], n], {n}], {n, 2, 100}];
u = Flatten[Partition[t, 2]];
v = Table[n (n + 1), {n, 1, 80}];
d = Delete[u, Map[{#} &, v]]; (* A349526 *)
p = Table[{d[[n]], d[[n + 1]]}, {n, 1, 150}];
q = Map[Total, p]  (* A349946 *)
r = Table[Flatten[Position[q, n]], {n, 2, 12}]  (* A349947 array *)
Flatten[r]  (* A349947 sequence *)

A329949 Lexicographically earliest sequence of positive numbers such that following proposition is true: a(n) is the number of occurrences of a(n+1) in the sequence so far, up to and including a(n+1).

Original entry on oeis.org

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

Offset: 1

Eric Angelini, Nov 25 2019

It is easy to see how regular the sequence is by transforming it into a triangle with 2k+1 terms in each row, where k runs through the natural numbers. Then one sees that a(k^2) = 1, a(k^2 + 1) = k+1 etc. - Ivan N. Ianakiev, Nov 26 2019

			a(1) = 1 means that there is 1 a(2) so far in the sequence - which is true, there is one "2" in the sequence up to a(2);
a(2) = 2 means that there are 2 a(3) so far in the sequence - which is true, there are two "2" in the sequence up to a(3);
a(3) = 2 means that there are 2 a(4) so far in the sequence - which is true, there are two "1" in the sequence up to a(4);
a(4) = 1 means that there is 1 a(4) so far in the sequence - which is true, there is one "3" in the sequence up to a(5);
a(5) = 3 means that there are 3 a(6) so far in the sequence - which is true, there are three "2" in the sequence up to a(6); etc.

Cf. A349526

cp's OEIS Frontend

A349946 a(n) = A349526(n) + A349526(n+1).

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A349947 Triangular array: row n gives the positions of n+1 in A349946.

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Programs

Mathematica

A329949 Lexicographically earliest sequence of positive numbers such that following proposition is true: a(n) is the number of occurrences of a(n+1) in the sequence so far, up to and including a(n+1).

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