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 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, ... 1
 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, 132, 25, 156, 27, 182, 14, 29, 210, 31, 240, 16, 33, 272, 35, 306, 18, 37, 342, 39, 380, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A permutation of the nonnegative integers. 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] 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). 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. A302906 is the sequence of starting indices of these sets. LINKS J. Stauduhar, permutation of nonnegative integers, SeqFan list, Apr 11 2018 EXAMPLE 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: (0,1) -> {0, 0+1, 0*1, 1} -> [0,1] (1,2) -> {1, 1+2, 1*2, 2} -> [0,1,3,2] (2,3) -> {2, 2+3, 2*3, 3} -> [0,1,3,2,5,6] (3,4) -> {3, 3+4, 3*4, 4} -> [0,1,3,2,5,6,7,12,4] etc. 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. PROG (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 CROSSREFS Cf. A000096, A002378, A024701. Sequence in context: A269376 A257793 A083236 * A277820 A277680 A303764 Adjacent sequences:  A302714 A302715 A302716 * A302718 A302719 A302720 KEYWORD nonn AUTHOR J. Stauduhar, Apr 12 2018 STATUS approved

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Last modified December 16 09:06 EST 2019. Contains 330020 sequences. (Running on oeis4.)