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A329544 Add the odd terms and subtract the even ones, the result must always be a palindrome. This is the lexicographically earliest sequence of distinct positive integers with this property.

%I A329544 #59 Mar 07 2020 11:17:11
%S A329544 1,3,2,5,4,19,11,22,6,17,14,8,7,15,16,27,24,13,18,29,26,37,33,44,28,
%T A329544 39,36,25,30,41,38,49,46,35,40,51,48,59,45,10,68,32,21,20,9,55,58,47,
%U A329544 50,61,60,71,66,77,23,12,88,191,101,111,91,112,31,81,121,131,141,70,132,80,122,90,142,174,43,54,72,83,53,42
%N A329544 Add the odd terms and subtract the even ones, the result must always be a palindrome. This is the lexicographically earliest sequence of distinct positive integers with this property.
%C A329544 Negative palindromes are not allowed (0 is OK). After 50000 terms, the smallest unused integers are 919, 1020, 1029, 1031, 1038, 1041, 1047, ... and the largest used is 208831. The largest palindrome produced so far is 1000001. Is the sequence a permutation of the integers > 0?
%C A329544 After one million terms, the smallest unused integers are still the seven mentioned (above) for 50000 terms. - _Hans Havermann_, Nov 27 2019
%C A329544 This sequence is not a permutation of the nonnegative integers because it cannot contain any term of A104444. The value 919 may only appear after a running total equal to 0 (see A083142, A084843). - _Rémy Sigrist_, Dec 11 2019. There are only two 0's in the first million terms of A329796, at n=12 and n=1002, so the chance that this happens seems slight. On the other hand, the zeros in the base 3 analog, A330314, are more plentiful (see A330325), so further investigation is needed. - _Hans Havermann_ and _N. J. A. Sloane_, Dec 12 2019
%H A329544 Jean-Marc Falcoz, <a href="/A329544/b329544.txt">Table of n, a(n) for n = 1..50000</a>.
%H A329544 Hans Havermann, <a href="/A329544/a329544.png">High-resolution graph of 100000 terms showing even/odd distribution</a>
%H A329544 N. J. A. Sloane, <a href="/A329544/a329544.txt">Table of n, a(n), A329796(n), A329796(n)/a(n) for n = 1..50000</a>
%H A329544 N. J. A. Sloane (in collaboration with Scott R. Shannon), <a href="/A331452/a331452.pdf">Art and Sequences</a>, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
%e A329544 The sequence starts with 1 which is positive and a palindrome.
%e A329544 1 + 3 = 4 (palindrome). (2 is not allowed because 1 - 2 < 0.)
%e A329544 1 + 3 - 2 = 2 (palindrome)
%e A329544 1 + 3 - 2 + 5 = 7 (palindrome)
%e A329544 1 + 3 - 2 + 5 - 4 = 3 (palindrome)
%e A329544 1 + 3 - 2 + 5 - 4 + 19 = 22 (palindrome)
%e A329544 1 + 3 - 2 + 5 - 4 + 19 + 11 = 33 (palindrome)
%e A329544 1 + 3 - 2 + 5 - 4 + 19 + 11 - 22 = 11 (palindrome), etc.
%o A329544 (PARI) A329544_vec(N,u=1,U,a,s,d)={vector(N,n, a=u; while(bittest(U,a-u)|| Vecrev(d=digits(s-(-1)^a*a))!=d|| (a>s&&!bittest(a,0)),a++); s-=(-1)^a*a; U+=1<<(a-u); while(bittest(U,0), U>>=1; u++);a)} \\ _M. F. Hasler_, Nov 16 2019
%Y A329544 Cf. A329545 (same idea, but where the odd integers are subtracted and the even ones are added).
%Y A329544 Cf. A002113 (palindromes), A086862 (first differences), A104444, A329796 (running totals), A329797, A329798 (records), A330311 (when n appears).
%Y A329544 See also A083142, A084843, A330313, A330314, A330325.
%K A329544 base,nonn,look
%O A329544 1,2
%A A329544 _Eric Angelini_ and _Jean-Marc Falcoz_, Nov 16 2019