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.
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, 28, 27, 30, 29, 32, 31, 35, 33, 36, 34, 37, 38, 42, 39, 43, 40, 44, 41, 45, 47, 46, 48, 55, 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
Offset: 1
Keywords
Examples
As a(1) = 2 has an odd index, the rook moves 2 terms to the Right on a(3) = 5; from there the rook moves according to a(2) = 1 (1 term to the L) on a(2) = 1; from there the rook moves according to a(3) = 5 (5 terms to the R) on a(7) = 7; from there the rook moves according to a(4) = 3 (3 terms to the L) on a(4) = 3; 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.
Links
- Eric Angelini, La tour d'échecs et l'essuie-glace, Cinquante signes, 2021.
- Eric Angelini, La tour d'échecs et l'essuie-glace, Cinquante signes, 2021. [Cached copy]
- Index entries for sequences that are permutations of the natural numbers
Crossrefs
Cf. A285471.
Programs
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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
Extensions
Edited and better definition from M. F. Hasler, Jul 19 2021
Comments