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