A384688 Runs of t in the range 0 <= t <= k and the same parity as k, for successive k >= 0.

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

Offset

0,4

Comments

The corresponding k is A055086(n), or k+1 = A000267(n).

A run is 0, 2, 4, ..., k when k even, or 1, 3, 5, ..., k when k odd, and has length floor(k/2) + 1.

Runs start at quarter squares n = A002620(k+1), with those beginning 0 at oblong numbers n = A002378(i) and those starting 1 at the squares n = (i+1)^2 (for i >= 0 in both cases).

Starts to differ from A025643 at n=109.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..5001

Formula

a(n) = 2*r+1 if r < s or a(n) = 2*(r-s) otherwise, where square root and remainder n = s^2 + r being s=A000196(n), r=A053186(n).

a(n) = ceiling(A053186(4*n+1) / 2).

a(n) = A055086(n) - 2*A216607(n+1).

a(n) = 2*A055087(n) + A079813(n+1).

Example

Runs and their corresponding k = A055086(n) begin,
  n          = 0  1  2    4    6      9
  a(n)       = 0, 1, 0,2, 1,3, 0,2,4, 1,3,5, ...
  A055086(n) = 0, 1, 2,2, 3,3, 4,4,4, 5,5,5, ...

Mathematica

ClearAll[a] a[n_Integer]:=Module[{s, r}, s=Floor[Sqrt[n]]; r=n-s^2; If[r<s, 2*r+1, 2*(r-s)]] Table[a[n], {n, 0, 100}] (* Vincenzo Librandi, Jul 06 2025 *)

Prog

(PARI) a(n) = my(r, s=sqrtint(n, &r)); if(r<s, r<<1+1, (r-s)<<1);

Crossrefs

Cf. A000196, A000267, A053186, A055086, A055087, A079813, A216607.

Cf. A002620, A002378 (indices of 0's), A000290 (indices of 1's).

Sequence in context: A025641 A025649 A025642 * A025643 A127478 A127472

Adjacent sequences: A384685 A384686 A384687 * A384689 A384690 A384691

Keyword

nonn,easy

Author

Kevin Ryde, Jun 07 2025

Status

approved