A279316 -id:A279316 - cp's OEIS frontend

A279313 Period 14 zigzag sequence: repeat [0,1,2,3,4,5,6,7,6,5,4,3,2,1].

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

Offset: 0

Wesley Ivan Hurt, Dec 09 2016

Decimal expansion of 1111111/90000009. - Elmo R. Oliveira, Feb 21 2024

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,-1,1).

Period k zigzag sequences: A000035 (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), this sequence (k=14), A279319 (k=16), A158289 (k=18).

Cf. A279316, A279321.

&cat[[0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1]: n in [0..10]];

A279313:=n->[0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1][(n mod 14)+1]: seq(A279313(n), n=0..200);

CoefficientList[Series[x*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)/(1 - x + x^7 - x^8), {x, 0, 100}], x]

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; 1,-1,0,0,0,0,0,1]^n*[0;1;2;3;4;5;6;7])[1,1] \\ Charles R Greathouse IV, Dec 12 2016

G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)/(1 - x + x^7 - x^8).
a(n) = a(n-1) - a(n-7) + a(n-8) for n > 7.
a(n) = abs(n - 14*round(n/14)).
a(n) = Sum_{i=1..n} (-1)^floor((i-1)/7).
a(2n) = 2*A279316(n), a(2n+1) = A279321(n).
a(n) = a(n-14) for n >= 14. - Wesley Ivan Hurt, Sep 07 2022

A053616 Pyramidal sequence: distance to nearest triangular number.

Original entry on oeis.org

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

Henry Bottomley, Mar 20 2000

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)

			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)

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

Cf. A000217, A002262, A053188, A172471.

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

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 *)

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

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

from math import isqrt
def A053616(n): return abs((m:=isqrt(k:=n<<1))*(m+1)-k)>>1 # Chai Wah Wu, Jul 15 2022

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
From Ctibor O. Zizka, Nov 12 2024: (Start)
For s >= 1, t from [0, s] :
a(2*s^2 + t) = s - t.
a(2*s^2 - t) = s - t.
a(2*s^2 + 2*s - t) = s - t.
a(2*s^2 + 2*s + 1 + t) = s - t. (End)

cp's OEIS Frontend

A279313 Period 14 zigzag sequence: repeat [0,1,2,3,4,5,6,7,6,5,4,3,2,1].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A053616 Pyramidal sequence: distance to nearest triangular number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula