A305258 - OEIS

A305258

List of y-coordinates of a point moving in a smooth counterclockwise spiral rotated by Pi/4.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Table of n, a(n) for n=0..81.

Hugo Pfoertner, Illustration of spiral.

Index entries for sequences related to coordinates of 2D curves

FORMULA

a(n) = A053616(n)*sign(sin(Pi*(1+sqrt(1+8*n))/2)), so that abs(a(n)) = A053616(n).

a(n) = A010751(n-floor((1/2)*(sqrt(2n-1)+1))). - William McCarty, Jul 29 2021

EXAMPLE

Sequence gives y-coordinate of the n-th point of the following spiral:

4 | 32 49

| / \ \

3 | 33 18 31 48

| / / \ \ \

2 | 34 19 8 17 30 47

| / / / \ \ \ \

1 | 35 20 9 2 7 16 29 46

| / / / / \ \ \ \ \

0 | 36 21 10 3 0---1 6 15 28 45

| \ \ \ \ / / / /

-1 | 37 22 11 4---5 14 27 44

| \ \ \ / / /

-2 | 38 23 12--13 26 43

| \ \ / /

-3 | 39 24--25 42

| \ /

-4 | 40--41

_______________________________________

x: -4 -3 -2 -1 0 1 2 3 4 5

PROG

(PARI) up=-1; print1(x=0, ", "); for(stride=1, 12, up=-up; x+=stride; y=x+stride+1; for(k=x, y-1, print1(up*min(k-x, y-k), ", "))) \\ Hugo Pfoertner, Jun 02 2018

CROSSREFS

A010751 gives sequence of x-coordinates.

Cf. A053616.

Cf. A000384, A001105, A002024, A014105, A046092.

Sequence in context: A170984 A114021 A239287 * A053616 A365746 A094718

Adjacent sequences: A305255 A305256 A305257 * A305259 A305260 A305261

KEYWORD

sign

AUTHOR

Hugo Pfoertner, May 29 2018

STATUS

approved