A135317 Let h(2*n, 1) = 2*n and h(2*n, m) = h(2*n, m-1) + 2 * d(2*n, m) for m > 1, where d(2*n, m) = (least multiple of m not less than h(2*n, m-1)) - h(2*n, m-1). Then d(2*n, m) is eventually 2-periodic as a function of m, and a(n) is defined as d(2*n, 2*m+1) for large m.

0, 1, 2, 0, 1, 2, 3, 4, 0, 3, 4, 5, 1, 2, 3, 6, 0, 1, 4, 5, 6, 2, 5, 6, 7, 8, 0, 7, 8, 9, 3, 4, 5, 1, 2, 3, 6, 7, 10, 4, 7, 8, 0, 1, 2, 9, 10, 11, 5, 8, 9, 12, 6, 7, 10, 11, 12, 8, 9, 10, 0, 3, 4, 13, 1, 2, 5, 6, 11, 3, 4, 9, 14, 0, 1, 10, 11, 12, 2, 7, 8, 13, 5, 6, 9, 12, 13, 7, 10, 11, 14, 15, 16, 14

Offset

0,3

Comments

Sequence yielding an ordering of N*N derived from a family of recurrences.

For any integer k define h(k,1)=k and for m>1 define h(k,m)=h(k,m-1)+2*((-h(k,m-1)) mod m) where "r mod s" denotes least nonnegative residue of r modulo s [informally, h(k,m) is got by "reflecting" h(k,m-1) in the least multiple of m that is >=h(k,m-1)]. Then for fixed k>=0 there are integers c(k), b(k), m(k) such that for all m>m(k) we have h(k,2*m+1)-h(k,2*m)=2*c(k) and h(k,2*m+2)-h(k,2*m+1)=2*b(k). [In the terms of the function d defined in the Name, c(k) = d(k, 2*m+1) and b(k) = d(k, 2*m+2) for all m>m(k).]

For all n we have c(2*n+1)=c(2*n) and b(2*n+1)=1+b(2*n). Moreover b(2*n) is even for all n. The function n->(c(2*n),b(2*n)/2) is a bijection from the nonnegative integers N to N*N. It is "monotone" in the sense that n<=n' whenever c(2*n)<=c(2*n') and b(2*n)<=b(2*n').

This sequence is a(n) = c(2*n).

The results on which the definition is based are not yet proved, but they are plausible and overwhelmingly supported by numerical evidence.

For each fixed m, k->h(k,m) is a bijection Z->Z (this is easy!). However for k<0 the sequence h(k,m) does not have the pseudo-periodic property we have used in defining c(k) and b(k).

m(k) appears to be O(sqrt k).

Links

Table of n, a(n) for n=0..93.

Formula

It appears that a(n) = A319573(A252448(2*n+1)). [discovered by Sequence Machine] Also, it appears that b(2*n) = A319572(A252448(2*n)), where b is described above. - Andrey Zabolotskiy, Apr 28 2023

Example

h(18,m) for m>=1 goes 18,18,18,22,28,32,38,42,48...so we can take m(18) = 2, then 2*c(18)=28-22=38-32=48-42=...=6, so a(9) = c(18) = 3 and similarly b(18) = 2.

Mathematica

d[h_, m_] := (Ceiling[#] - # &[h/m]) m;
a[n_] := Module[{h = 2 n, b, c, m = 1},
   While[m <= Max[5, n], (*condition is conjectural*)
    m++; b = d[h, m]; h += 2 b;
    m++; c = d[h, m]; h += 2 c];
   c];
Table[a[n], {n, 0, 93}]; (* Andrey Zabolotskiy, Apr 29 2023 *)

Crossrefs

Cf. A319571, A319572, A319573, A252448.

Sequence in context: A350603 A262678 A238794 * A332663 A227179 A115218

Adjacent sequences: A135314 A135315 A135316 * A135318 A135319 A135320

Keyword

nonn

Author

Alex Abercrombie, Feb 15 2008

Extensions

Edited by Andrey Zabolotskiy, Apr 29 2023

Status

approved