cp's OEIS Frontend

A302717 Start with a(0) = 0, then append the terms in [x, 2*x+1, x*(x+1)] which do not occur earlier, for x = 1, 2, ...

%I A302717 #44 Dec 23 2024 14:53:45
%S A302717 0,1,3,2,5,6,7,12,4,9,20,11,30,13,42,15,56,8,17,72,19,90,10,21,110,23,
%T A302717 132,25,156,27,182,14,29,210,31,240,16,33,272,35,306,18,37,342,39,380,
%U A302717 41,420,43,462,22,45,506,47,552,24,49,600,51,650,26,53,702,55,756,28,57,812,59,870,61,930,63,992,32,65,1056,67,1122,34,69,1190
%N A302717 Start with a(0) = 0, then append the terms in [x, 2*x+1, x*(x+1)] which do not occur earlier, for x = 1, 2, ...
%C A302717 A permutation of the nonnegative integers.
%C A302717 If a(n) is in A024701 (i.e., of the form (prime^2-1)/4), then a(n-1) is prime. Indeed, A024701(m) = k*(k+1) with k = (prime(m+1)-1)/2, and any term k*(k+1) > 0 is preceded by 2*k+1 = prime(m+1). [Edited and proof added by _M. F. Hasler_, Apr 13 2018]
%C A302717 The term x*(x+1) will always be appended since it is larger than all preceding terms (except for x = 1), and also 2*x+1 cannot occur earlier because it is odd while x*(x+1) is always even. So only the term x will be inserted (or not) in a somewhat irregular pattern, namely whenever x is an even but not oblong number (A002378). We see that this is the case for x = 4, 8, 10, 14, 16, 18, 22, ...; recognizable by the fact that a(n) = (a(n+1)-1)/2 and equivalently, there are two and not only one smaller number between two larger "records" x*(x+1).
%C A302717 If we count the terms added from each 4-tuple during each iteration we find that either two or three terms are added: 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, ... where the set of three twos (2, 2, 2) appears with decreasing frequency.
%C A302717 A302906 is the sequence of starting indices of these sets.
%H A302717 J. Stauduhar, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2018-April/018577.html">permutation of nonnegative integers</a>, SeqFan list, Apr 11 2018
%H A302717 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A302717 Repeatedly take consecutive numbers a and b and append to the sequence any of {a, a+b, a*b, b} not already in the sequence. Beginning with a=0 and b=1:
%e A302717 (0,1) -> {0, 0+1, 0*1, 1} -> [0,1]
%e A302717 (1,2) -> {1, 1+2, 1*2, 2} -> [0,1,3,2]
%e A302717 (2,3) -> {2, 2+3, 2*3, 3} -> [0,1,3,2,5,6]
%e A302717 (3,4) -> {3, 3+4, 3*4, 4} -> [0,1,3,2,5,6,7,12,4]
%e A302717 etc.
%e A302717 In the above construction, we always have b = a+1. Thus [a, a+b, a*b, b] = [a, 2*a+1, a*(a+1), a+1], and a simpler description is to consider only { a, 2*a+1, a*(a+1) }, the 4th term being equal to the 1st term of the next 4-tuple. To ensure we have a permutation of the integers >= 0 starting at index 0 and not a list stating at index 1, we can fix a(0) = 0 explicitly and then go on with a = x = 1, 2, 3, ... to get the same sequence.
%o A302717 (PARI) u=[];(do(x)=setsearch(u,x)||print1(x",")||u=setunion(u,[x]));for(a=0,199,do(a);do(2*a+1);do(a^2+a)) \\ _M. F. Hasler_, Apr 12 2018
%Y A302717 Cf. A000096, A002378, A024701.
%K A302717 nonn
%O A302717 0,3
%A A302717 _J. Stauduhar_, Apr 12 2018