A053616 Pyramidal sequence: distance to nearest triangular number.

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1

Offset

0,9

Comments

From Wolfdieter Lang, Oct 24 2020: (Start)

If this sequence is written with offset 1 as a number triangle T(n, k), with n the length of row n, for n >= 1, then row n gives the primitive period of the periodic sequence {k (mod* n)}_{k>=0}, where k (mod* n) = k (mod n) if k <= floor(n/2) and otherwise it is -k (mod n). Such a modified modular relation mod* n has been used by Brändli and Beyne, but for integers relative prime to n.

These periodic sequences are given in A000007, A000035, A011655, A007877, |A117444|, A260686, A279316, for n = 1, 2, ..., 7. For n = 10 A271751, n = 12 A271832, n = 14 A279313. (End)

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2015-2016.

Index entries for sequences related to distance to nearest element of some set

Formula

a(n) = (x - |y - |x-y||)/2, when (x,y) is the n-th element in the triangle x >= y >= 1. - M. F. Hasler, Dec 06 2019

a(n) = (1/2)*abs(t^2 + t - 2*n), where t = floor(sqrt(2*n)) = A172471. - Ridouane Oudra, Dec 15 2021

Example

a(12) = |12 - 10| = 2 since 10 is the nearest triangular number to 12.
From M. F. Hasler, Dec 06 2019: (Start)
Ignoring a(0) = 0, the sequence can be written as triangle indexed by m >= k >= 1, in which case the terms are (m - |k - |m-k||)/2, as follows:
   0,      (Row 0: ignore)
   0,      (Row m=1, k=1: For k=m, m - |k - |m-k|| = m - |m - 0| = 0.)
   1, 0,        (Row m=2: for k=1, |m-k| = 1, k-|m-k| = 0, m-0 = 2, (...)/2 = 1.)
   1, 1, 0,
   1, 2, 1, 0,    (Row m=4: for k=2, we have twice the value of (m=2, k=1) => 2.)
   1, 2, 2, 1, 0,
   (...)
This is related to the non-associative operation A049581(x,y) = |x - y| =: x @ y. Specifically, @ is commutative and any x is its own inverse, so non-associativity of @ can be measured through the commutator ((x @ y) @ y) @ x which equals twice the element indexed {m,k} = {x,y} in the above triangle.
(End)

Mathematica

a[n_] := (k =.; k = Reduce[k > 0 && k*(k+1)/2 == n, Reals][[2]] // Floor; Min[(k+1)*(k+2)/2 - n, n - k*(k+1)/2]); Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Jan 08 2013 *)
Module[{trms=120, t}, t=Accumulate[Range[Ceiling[(Sqrt[8*trms+1]-1)/2]]]; Join[{0}, Flatten[Table[Abs[Nearest[t, n][[1]]-n], {n, trms}]]]] (* Harvey P. Dale, Nov 08 2013 *)

Prog

(PARI) print1(x=0, ", "); for(stride=1, 13, x+=stride; y=x+stride+1; for(k=x, y-1, print1(min(k-x, y-k), ", "))) \\ Hugo Pfoertner, Jun 02 2018
(PARI) apply( {a(n)=if(n, -abs(n*2-(n=sqrtint(8*n-7)\/2)^2)+n)\2}, [0..40]) \\ same as (i - |j - |i-j||)/2 with i=sqrtint(8*n-7)\/2, j=n-i(i-1)/2. - M. F. Hasler, Dec 06 2019
(Python)
from math import isqrt
def A053616(n): return abs((m:=isqrt(k:=n<<1))*(m+1)-k)>>1 # Chai Wah Wu, Jul 15 2022

Crossrefs

Cf. A000217, A002262, A053188, A172471.

a(n) = abs(A305258(n)).

Sequence in context: A114021 A239287 A305258 * A365746 A094718 A076191

Adjacent sequences: A053613 A053614 A053615 * A053617 A053618 A053619

Keyword

easy,nice,nonn

Author

Henry Bottomley, Mar 20 2000

Status

approved