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A175428 a(n) is the smallest positive integer such that a(n)+a(n-1) is a cube and a(n) is not in {a(1), ..., a(n-1)}.

%I A175428 #24 Jun 02 2015 15:40:51
%S A175428 1,7,20,44,81,135,208,8,19,45,80,136,207,9,18,46,79,137,206,10,17,47,
%T A175428 78,138,205,11,16,48,77,139,204,12,15,49,76,140,203,13,14,50,75,141,
%U A175428 202,310,33,31,94,122,3,5,22,42,83,133,210,6,2,25,39,86,130,213,299,430,82
%N A175428 a(n) is the smallest positive integer such that a(n)+a(n-1) is a cube and a(n) is not in {a(1), ..., a(n-1)}.
%C A175428 It appears that this is a permutation of the positive integers. 4 occurs as a(149).
%C A175428 See A034175 for the case where a(n)+a(n-1) is a square.
%C A175428 The parity of a(n) equals A133872(n) until n=56 after which there are occasional deviations from the pattern. This is clearly shown by considering the run lengths of the parity for this sequence: 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,2,1,2,3,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,1,2,2,2,1,2,2,3,2,1,2,2,2,3,2,2,2,2,2,2,1,2,2,2,1,2,3,2,2,2,3,2,2,1,2,2,1,... - _Jeremy Gardiner_, Feb 16 2014
%H A175428 Jeremy Gardiner, <a href="/A175428/b175428.txt">Table of n, a(n) for n = 1..1319</a>
%t A175428 a = {1}; For[i = 2, i < 66, i++, k = 1; While[! IntegerQ@ Power[k + a[[i - 1]] && ! MemberQ[a, k], 1/3], k++]; AppendTo[a, k]]; a (* _Michael De Vlieger_, Jun 02 2015 *)
%o A175428 (PARI) v=[1];n=1;while(n<10^3,if(ispower((v[#v]+n),3)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0);n++);v \\ _Derek Orr_, Jun 01 2015
%Y A175428 Cf. A034175, A011934, A000578 (the cubes), A133872.
%K A175428 nonn,easy
%O A175428 1,2
%A A175428 _John W. Layman_, May 10 2010