A343611 Partial sums of A345967.

0, 2, 1, 6, 3, 9, 5, 12, 4, 16, 7, 18, 8, 23, 10, 27, 13, 29, 11, 33, 14, 35, 15, 40, 17, 43, 19, 47, 20, 50, 21, 53, 22, 57, 24, 60, 26, 63, 25, 67, 28, 71, 31, 75, 34, 79, 32, 78, 30, 85, 36, 87, 37, 90, 38, 95, 41, 97, 39, 101, 42, 105, 45, 109, 48, 113, 46, 112, 44, 118, 49, 121, 51

Offset

0,2

Comments

Explicitly, this is the sequence of distinct nonnegative integers such that the sequence of first differences d(n) = a(n+1) - a(n) has alternating signs, distinct nonnegative partial sums, and the absolute values are all distinct and the lexicographically earliest sequence formed that way.

The first differences are (2, -1, 5, -3, 6, -4, 7, -8, 12, -9, 11, -10, 15, ...). Taking absolute values yields sequence S = A345967 which is the lexico-earliest sequence of distinct positive integers such that the alternating partial sums (i.e., a(n) = Sum_{k=1..n} -(-1)^k S(k), n >= 0) give all nonnegative integers exactly once.

Links

Table of n, a(n) for n=0..72.

Index entries for sequences that are permutations of the natural numbers

Formula

a(n) = Sum_{k=1..n} -(-1)^n * A345967(k), n >= 0.

Prog

(PARI) A343611_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, US); break); P)} \\ Gives the vector a(1..Nmax), i.e., without a(0)=0.

Crossrefs

Cf. A345967.

Sequence in context: A285355 A316605 A307178 * A026191 A338206 A050137

Adjacent sequences: A343608 A343609 A343610 * A343612 A343613 A343614

Keyword

nonn

Author

M. F. Hasler, Jul 11 2021

Status

approved