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%I #35 Jun 27 2022 21:18:41
%S 1,2,1,3,2,4,5,3,6,7,4,8,5,9,10,6,11,7,12,13,8,14,15,9,16,10,17,18,11,
%T 19,20,12,21,13,22,23,14,24,15,25,26,16,27,28,17,29,18,30,31,19,32,20,
%U 33,34,21,35,36,22,37,23,38,39,24,40,41,25
%N a(n) = smallest k such that k=a(n-k-1) is the only appearance of k so far; if there is no such k, then a(n) = least positive integer that has not yet appeared.
%C From _Daniel Joyce_, Apr 13 2001: (Start)
%C This sequence displays every positive integer exactly twice and the gap between the two occurrences of n contains exactly n other values. The first occurrence of n precedes the first occurrence of n+1.
%C Also related to the Wythoff array (A035513) and the Para-Fibonacci sequence (A035513) where every positive integer is displayed exactly once in the whole array. Take any integer n in A026272 and let C = number of terms from the beginning of the sequence to the second occurrence of n. Then C = (2nd term after n in the applicable sequence for n in A035513).
%C Also in the second occurrence of n in A026272, let N=n ( - one term) = (first term value after n in the applicable sequence for n in A035513). In this format the second occurrence of n in A026272 will produce in A035513, n itself and two of the succeeding terms of n in the Wythoff array where every positive integer can only be displayed once.
%C In A026272 if |a(n)-a(n+1)| > 10 then phi ~ a(n)/|a(n)-a(n+1)|. When n -> infinity it will converge to phi. (End)
%C Or, put a copy of n in A000027 n places further along! - _Zak Seidov_, May 24 2008
%C Another version would prefix this sequence with two leading 0's (see the Angelini reference). If we use this form and write down the indices of the two 0's, the two 1's, the two 2's, the two 3's, etc., then we get A072061. - _Jacques ALARDET_, Jul 26 2008
%D E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
%H Zak Seidov, <a href="/A026272/b026272.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A026242(n+2) - 1 = A026350(n+3) - 2 = A026354(n+4) - 3.
%t s=Range[1000];n=0;Do[n++;s=Insert[s,n,Position[s,n][[1]]+n+1],{500}];A026272=Take[s,1000] (* _Zak Seidov_, May 24 2008 *)
%o (PARI) A026272=apply(t->t-1,A026242[3..-1]) \\ Use vecextract(A026242,"3..") in PARI versions < 2.7. - _M. F. Hasler_, Sep 17 2014
%o (Python)
%o from collections import Counter
%o from itertools import count, islice
%o def agen(): # generator of terms
%o aset, alst, k, mink, counts = set(), [0], 0, 1, Counter()
%o for n in count(1):
%o for k in range(1, len(alst)-1):
%o if k == alst[n-k-1] and counts[alst[n-k-1]] == 1: an = k; break
%o else: an = mink
%o yield an; aset.add(an); alst.append(an); counts.update([an])
%o while mink in aset: mink += 1
%o print(list(islice(agen(), 66))) # _Michael S. Branicky_, Jun 27 2022
%Y Cf. A000027, A035513, A014552, A176127.
%K nonn,easy,nice
%O 1,2
%A _Clark Kimberling_
%E Edited by _Max Alekseyev_, May 31 2011
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