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A345967 Lexicographically first sequence of distinct positive integers such that the alternating partial sums p(n) = Sum_{k=1..n} -(-1)^k a(k), n >= 1, are distinct positive integers.

%I A345967 #28 Dec 21 2024 20:22:05
%S A345967 2,1,5,3,6,4,7,8,12,9,11,10,15,13,17,14,16,18,22,19,21,20,25,23,26,24,
%T A345967 28,27,30,29,32,31,35,33,36,34,37,38,42,39,43,40,44,41,45,47,46,48,55,
%U A345967 49,51,50,53,52,57,54,56,58,62,59,63,60,64,61,65,67,66,68,74,69,72,70,75,71,73,76,79,77,80,78
%N A345967 Lexicographically first sequence of distinct positive integers such that the alternating partial sums p(n) = Sum_{k=1..n} -(-1)^k a(k), n >= 1, are distinct positive integers.
%C A345967 The chess rook as a windshield wiper sequence: terms with an odd index [a(1), a(3), a(5), ...] move the chess rook horizontally to the right over a(n) terms; terms with an even index [a(2), a(4), a(6), ...] move the chess rook to the left over a(n) terms; this is the lexicographically earliest sequence of positive distinct terms such that all terms of the sequence will be visited exactly once by the rook.
%C A345967 It turns out that both, sequence (a(n), n >= 1) and that of partial alternating sums (p(n), n >= 1), are permutations of the positive integers. - _M. F. Hasler_, Jul 11 2021
%C A345967 The inverse permutation of this sequence starts (2, 1, 4, 6, 3, 5, 7, 8, 10, 12, 11, 9, 14, 16, 13, 17, 15, 18, 20, 22, 21, ...). - _M. F. Hasler_, Jul 19 2021
%H A345967 Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2021/06/la-tour-dechecs-et-lessuie-glace.html">La tour d'échecs et l'essuie-glace</a>, Cinquante signes, 2021.
%H A345967 Eric Angelini, <a href="/A345967/a345967.pdf">La tour d'échecs et l'essuie-glace</a>, Cinquante signes, 2021. [Cached copy]
%H A345967 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A345967 As a(1) = 2 has an odd index, the rook moves 2 terms to the Right on a(3) = 5;
%e A345967 from there the rook moves according to a(2) = 1 (1 term to the L) on a(2) = 1;
%e A345967 from there the rook moves according to a(3) = 5 (5 terms to the R) on a(7) = 7;
%e A345967 from there the rook moves according to a(4) = 3 (3 terms to the L) on a(4) = 3;
%e A345967 from there the rook moves according to a(5) = 6 (6 terms to the R) on a(10) = 9; etc. The rook's successive movements can be seen as the movements of a windshield wiper.
%o A345967 (PARI) A345967_vec(Nmax, P=0)={ my(US=[0], UP=[P], used(x,U)= setsearch(U,x) || x<=U[1], insert(x,U)= U=setunion(U,[x]); while(#U>1&&U[2]==U[1]+1, U=U[^1]); U); vector(Nmax, n, my(s=(-1)^n); for(S=US[1]+1,oo, (used(S,US) || used(P-s*S,UP))&&next; if(s<0, my(f=1); for(PP=UP[1]+1,P+S-1, used(PP,UP) || used(P+S-PP,US) || PP==P || [f=0; break]); f && next); UP=insert(P-=s*S, UP); US=insert(s=S, US); break); s)} \\ _M. F. Hasler_, Jul 11 2021
%Y A345967 Cf. A285471.
%K A345967 nonn
%O A345967 1,1
%A A345967 _Eric Angelini_ and _Neil Bickford_, Jun 30 2021
%E A345967 Edited and better definition from _M. F. Hasler_, Jul 19 2021