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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.

%I A117384 #43 Sep 10 2021 06:04:49
%S A117384 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,
%T A117384 18,13,19,14,20,15,21,22,16,23,17,24,18,25,19,26,20,27,21,28,29,22,30,
%U A117384 23,31,24,32,25,33,26,34,27,35,28,36,37,29,38,30,39,31,40,32,41,33,42
%N 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.
%C A117384 Positions where n occurs are A001614(n) and 4*n-A001614(n), where A001614 is the Connell sequence: 1 odd, 2 even, 3 odd, ...
%C A117384 From _Paolo Xausa_, Aug 27 2021: (Start)
%C A117384 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).
%C A117384 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).
%C A117384 The triangle has the following properties.
%C A117384 Row lengths are the positive odd numbers.
%C A117384 First column is A000124.
%C A117384 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.
%C A117384 Row records (the positive terms of A000217) are in the right border.
%C A117384 Indices of row records are the positive terms of A000290.
%C A117384 Each row r contains r terms that are duplicated in the next row.
%C A117384 In each row, the sum of terms which are not already listed in the sequence gives the positive terms of A006003.
%C A117384 Row sums give A063488.
%C A117384 For rows r >= 2, row product is A057003(r)*A057003(r-1). (End)
%F A117384 a(4*a(n)-n) = a(n).
%F A117384 Lim_{n->infinity} a(n)/n = 1/2.
%F A117384 Lim_{n->infinity} (a(n+1)-a(n))/sqrt(n) = 1.
%F A117384 a( A001614(n) ) = n; a( 4n - A001614(n) ) = n.
%F A117384 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
%e A117384 9 first appears at position: A001614(9) = 14;
%e A117384 9 next appears at position: 4*9 - A001614(9) = 22.
%e A117384 From _Paolo Xausa_, Aug 27 2021: (Start)
%e A117384 Written as an irregular triangle T(r,c) the sequence begins:
%e A117384   r\c  1   2   3   4   5   6   7   8   9  10  11  12  13
%e A117384   1:   1;
%e A117384   2:   2,  1,  3;
%e A117384   3:   4,  2,  5,  3,  6;
%e A117384   4:   7,  4,  8,  5,  9,  6, 10;
%e A117384   5:  11,  7, 12,  8, 13,  9, 14, 10, 15;
%e A117384   6:  16, 11, 17, 12, 18, 13, 19, 14, 20, 15, 21;
%e A117384   7:  22, 16, 23, 17, 24, 18, 25, 19, 26, 20, 27, 21, 28;
%e A117384   ...
%e A117384 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.
%e A117384                1
%e A117384             2  1  3
%e A117384          4  2  5  3  6
%e A117384       7  4  8  5  9  6 10
%e A117384   11  7 12  8 13  9 14 10 15
%e A117384              ...
%e A117384 (End)
%t A117384 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 *)
%t A117384 (* Second program, triangle rows *)
%t A117384 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 *)
%o A117384 (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]}
%o A117384 (PARI) T(r,c) = my(k = r-1-((c+1) % 2)); k*(k+1)/2+ceil(c/2);
%o A117384 tabf(nn) = {for (r=1, nn, for(c = 1, 2*r-1, print1(T(r,c), ", ");); print;);} \\ _Michel Marcus_, Sep 09 2021
%Y A117384 Cf. A117385 (a(5*a(n)-n)=a(n)), A117386 (a(6*a(n)-n)=a(n)).
%Y A117384 Cf. A001614 (Connell sequence).
%Y A117384 Cf. A000124, A000217, A006003, A057003, A063488, A344482.
%K A117384 nonn,tabf
%O A117384 1,2
%A A117384 _Paul D. Hanna_, Mar 11 2006