cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A343611 Partial sums of A345967.

Original entry on oeis.org

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

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Author

M. F. Hasler, Jul 11 2021

Keywords

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

Crossrefs

Cf. A345967.

Programs

  • 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.

Formula

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