A117384 Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 4*a(n), starting with a(1)=1 and filling the next vacant position with the smallest unused number.

1, 2, 1, 3, 4, 2, 5, 3, 6, 7, 4, 8, 5, 9, 6, 10, 11, 7, 12, 8, 13, 9, 14, 10, 15, 16, 11, 17, 12, 18, 13, 19, 14, 20, 15, 21, 22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28, 29, 22, 30, 23, 31, 24, 32, 25, 33, 26, 34, 27, 35, 28, 36, 37, 29, 38, 30, 39, 31, 40, 32, 41, 33, 42

Offset

1,2

Comments

Positions where n occurs are A001614(n) and 4*n-A001614(n), where A001614 is the Connell sequence: 1 odd, 2 even, 3 odd, ...

From Paolo Xausa, Aug 27 2021: (Start)

Terms can be arranged in an irregular triangle T(r,c) read by rows in which row r is a permutation P of the integers in the interval [s, s+rlen-1], where s = 1+(r-1)*(r-2)/2, rlen = 2*r-1 and r >= 1 (see example).

P is the alternating (first term > second term < third term > fourth term ...) permutation m -> 1, 1 -> 2, m+1 -> 3, 2 -> 4, m+2 -> 5, 3 -> 6, ..., rlen -> rlen, where m = ceiling(rlen/2).

The triangle has the following properties.

Row lengths are the positive odd numbers.

First column is A000124.

Terms in column c (where c >= 1) are of the form k*(k+1)/2+ceiling(c/2), for integers k >= floor((c-1)/2), each even column being equal to the column preceding it.

Row records (the positive terms of A000217) are in the right border.

Indices of row records are the positive terms of A000290.

Each row r contains r terms that are duplicated in the next row.

In each row, the sum of terms which are not already listed in the sequence gives the positive terms of A006003.

Row sums give A063488.

For rows r >= 2, row product is A057003(r)*A057003(r-1). (End)

Links

Table of n, a(n) for n=1..75.

Formula

a(4*a(n)-n) = a(n).

Lim_{n->infinity} a(n)/n = 1/2.

Lim_{n->infinity} (a(n+1)-a(n))/sqrt(n) = 1.

a( A001614(n) ) = n; a( 4n - A001614(n) ) = n.

T(r,c) = k*(k+1)/2+ceiling(c/2), where k = r-1-((c+1) mod 2), r >= 1 and c >= 1. - Paolo Xausa, Sep 09 2021

Example

9 first appears at position: A001614(9) = 14;
9 next appears at position: 4*9 - A001614(9) = 22.
From Paolo Xausa, Aug 27 2021: (Start)
Written as an irregular triangle T(r,c) the sequence begins:
  r\c  1   2   3   4   5   6   7   8   9  10  11  12  13
  1:   1;
  2:   2,  1,  3;
  3:   4,  2,  5,  3,  6;
  4:   7,  4,  8,  5,  9,  6, 10;
  5:  11,  7, 12,  8, 13,  9, 14, 10, 15;
  6:  16, 11, 17, 12, 18, 13, 19, 14, 20, 15, 21;
  7:  22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28;
  ...
The triangle can be arranged as shown below so that, in every row, each odd position term is equal to the term immediately below it.
               1
            2  1  3
         4  2  5  3  6
      7  4  8  5  9  6 10
  11  7 12  8 13  9 14 10 15
             ...
(End)

Mathematica

nterms=64; a=ConstantArray[0, nterms]; For[n=1; t=1, n<=nterms, n++, If[a[[n]]==0, a[[n]]=t; If[(d=4t-n)<=nterms, a[[d]]=a[[n]]]; t++]]; a (* Paolo Xausa, Aug 27 2021 *)
(* Second program, triangle rows *)
nrows = 8; Table[rlen=2r-1; Permute[Range[s=1+(r-1)(r-2)/2, s+rlen-1], Join[Range[2, rlen, 2], Range[1, rlen, 2]]], {r, nrows}] (* Paolo Xausa, Aug 27 2021 *)

Prog

(PARI) {a(n)=local(A=vector(n), m=1); for(k=1, n, if(A[k]==0, A[k]=m; if(4*m-k<=#A, A[4*m-k]=m); m+=1)); A[n]}
(PARI) T(r, c) = my(k = r-1-((c+1) % 2)); k*(k+1)/2+ceil(c/2);
tabf(nn) = {for (r=1, nn, for(c = 1, 2*r-1, print1(T(r, c), ", "); ); print; ); } \\ Michel Marcus, Sep 09 2021

Crossrefs

Cf. A117385 (a(5*a(n)-n)=a(n)), A117386 (a(6*a(n)-n)=a(n)).

Cf. A001614 (Connell sequence).

Cf. A000124, A000217, A006003, A057003, A063488, A344482.

Sequence in context: A120874 A358103 A112382 * A358799 A125160 A359027

Adjacent sequences: A117381 A117382 A117383 * A117385 A117386 A117387

Keyword

nonn,tabf

Author

Paul D. Hanna, Mar 11 2006

Status

approved